Datasets:

saier

/

unarXive_citrec

like 7

237e1229-3682-475d-898c-7619607f3c7c

Each gene was divided into 200 bp bins and levels of pol-II activity were computed at each time point as the total weighted count of reads overlapping each bin, where each read was weighted by how many basepairs in the read overlap the bin, as follows. Only uniquely mapped reads were used. For any read that at least partially overlaps a bin, the number of basepairs overlapping the bin was added into the activity level of the bin. For any read spanning multiple bins, the number of basepairs overlapping each bin were added into activity of that bin. The Genomatix mapping software provides alignment scores (values between 0 and 1; with our threshold only between 0.92 and 1) for mapping reads to the genome; for any read having alignment score less than 1, the number of overlapping basepairs added to each bin was multiplied by the alignment score. A noise removal was then done: a noise level was computed as the average activity level in 74 manually selected regions from Chromosome 1 that were visually determined to be inactive over the measurement time points, as follows. The regions were divided into 200 bp bins, and total weighted counts of reads overlapping each bin (each read weighted by the number of basepairs overlapping the bin) were computed in the same way as for the genes in the previous step. For each time point, the noise level was computed as the average activity level over all bins from all 74 regions. The computed noise level was subtracted from the mean of each bin in each gene, thresholding the result at zero. A list of the empty regions used is included as a Supplementary Dataset S1. As the number of ChIP-seq reads collected overall for pol-II varies between time points, a robust normalisation was done. After the previous noise removal step, for each gene \(g\) at each time \(t\) we compute the mean of the remaining activity (activity level after noise removal) over bins of the gene, denoted as \(r_{gt}\) . The activity levels are weighted counts of basepairs from reads overlapping the gene; we select genes having sufficient activity, that is, at least \(5\cdot 200\) overlapping basepairs from reads over each 200 bp bin of the gene, on average over the bins. For each gene \(g\) let \(T_{g} = \lbrace t^{\prime }\in \lbrace 5,10,\ldots ,1280\mathrm {\;min}\rbrace | r_{gt^{\prime }}>5\cdot 200\rbrace \) denote those time points (except the first time point) where the gene has sufficient activity. For each time point we compute a normalisation factor of [1]} \(C_t = \textrm {Median}_g \left\lbrace \frac{r_{gt}}{\textrm {GeomMean}_{t^{\prime }} \lbrace r_{gt^{\prime }}\rbrace } \right\rbrace \;.\) where \(\textrm {Median}_g\lbrace \cdot \rbrace \) denotes median over genes and\(\textrm {GeomMean}_{t^{\prime }}\lbrace r_{gt^{\prime }}\rbrace =(\prod _{t^{\prime }\in T_g} r_{gt^{\prime }})^{1/|T_g|}\) is the geometric mean over the time points having sufficient activity for gene \(g\) . The median is computed for time points after the first time point; for the first time point \(t=0\mathrm {\;min}\) we set \(C_t=1\) . The factor \(C_t\) normalises all the gene activity levels (weighted read counts) at a time point downwards if genes at that time point have unusually many reads, exceeding their (geometric) mean activity level, and normalises upwards if gene activity levels fall under their mean activity level. Lastly, time series summaries were computed for pol-II at each gene. For each gene at each time point \(t\) , the mean activity level (weighted read-count) of pol-II over bins in the 20% section of the gene nearest to transcription end was computed, normalised by \(C_t\) . This measured pol-II level represents transcriptional activity that had successfully passed through the gene to the transcription end site; it is expected to correspond better with mRNA production rate than pol-II activity at the transcription start of the gene, since pol-II near the transcription start site can be in the active or inactive state and after activation may require a significant time for transcription to complete. For a small number of genes where the active mRNA transcripts covered only part of the gene, we considered the area from the first active exon to last active exon, and summarised the gene using the 20% section nearest to the end of the area. Active transcripts were defined here as transcripts with a mean of more than 1.1 assigned counts in the BitSeq posterior expression estimates. BitSeq uses a prior that assigns 1 “pseudo-count” per transcript, so the active transcripts were only required to have minimal posterior expression that was distinguishable from the prior. A list of active transcripts is included as Supplementary Dataset S2. Lastly, for mathematical convenience, for each pol-II time series we subtracted from all time points the minimum value over the time points.

[1]

https://openalex.org/W2152239989

a6bdb79e-1cab-41de-a84a-f73542f3c8c5

The linear differential equation (REF ) and its linear solution operator (REF ) are similar to those used previously in [1]}, [2]}, [3]} except for the added delay. As in the previous works, the linearity of the solution operator permits exact joint Gaussian process (GP) modelling over \(p(t)\) and \(m(t)\) .

[1]

https://openalex.org/W2124584833

f73e05ef-d9a3-413d-8b76-5ef4f41cc0a1

The linear differential equation (REF ) and its linear solution operator (REF ) are similar to those used previously in [1]}, [2]}, [3]} except for the added delay. As in the previous works, the linearity of the solution operator permits exact joint Gaussian process (GP) modelling over \(p(t)\) and \(m(t)\) .

[2]

https://openalex.org/W2125373697

543533bc-59d5-4c4a-958d-3505de184843

For each gene, we model the pol-II activity time series in a nonparametric fashion by applying a GP prior over the shapes of the time series. Previous similar GP models [1]}, [2]}, [3]} have used a squared exponential covariance function for \(p(t)\) , as that allows derivation of all the shared covariances in closed form. This covariance has the limitation that it is stationary, and functions following it revert to zero away from data. These properties severely degrade its performance on our highly unevenly sampled data. To avoid this, we model \(p(t)\) as an integral of a function having a GP prior with a squared exponential covariance: then the posterior mean of \(p(t)\) tends to remain constant between observed data. That is, we model \(p(t)=p_0 + \int _{u=0}^t v(t) \mathrm {d}t\)

[1]

https://openalex.org/W2124584833

76bb2ed1-5562-413d-81a8-243794401c5f

For each gene, we model the pol-II activity time series in a nonparametric fashion by applying a GP prior over the shapes of the time series. Previous similar GP models [1]}, [2]}, [3]} have used a squared exponential covariance function for \(p(t)\) , as that allows derivation of all the shared covariances in closed form. This covariance has the limitation that it is stationary, and functions following it revert to zero away from data. These properties severely degrade its performance on our highly unevenly sampled data. To avoid this, we model \(p(t)\) as an integral of a function having a GP prior with a squared exponential covariance: then the posterior mean of \(p(t)\) tends to remain constant between observed data. That is, we model \(p(t)=p_0 + \int _{u=0}^t v(t) \mathrm {d}t\)

[2]

https://openalex.org/W2125373697

3b34371a-bd20-4124-8349-f89dd79433ff

Given the data and the priors for the parameters, we apply fully Bayesian inference with Hamiltonian Monte Carlo (HMC) sampling [1]} to obtain samples from the posterior distribution of the parameters. HMC is a MCMC algorithm that uses gradients of the target distribution to simulate a Hamiltonian dynamical system with an energy function based on the target distribution. This allows taking long steps while maintaining a high acceptance rate in the sampling.

[1]

https://openalex.org/W2059448777

7a96a816-66c8-4af1-8f08-63bfb90d563a

We run 4 parallel chains starting from different random initial states to allow convergence checking. We use the HMC implementation from NETLAB toolkit in Matlab with momentum persistence and number of leap frog steps \(\tau = 20\) which were found to work well in all cases. The step length \(\epsilon \) is tuned separately for every model (see below). After tuning, each chain is run for 10000 iterations. The samples are then thinned by a factor of 10, and the first half of the samples are discarded, leaving 500 samples from each chain, 2000 in all. Convergence is monitored using the potential scale reduction factor \(\sqrt{\hat{R}}\)  [1]}. \(\sqrt{\hat{R}}\) is computed separately for each variable, and if any of them is greater than 1.2, the result is discarded and a new sample obtained in a similar manner. The 9 genes that did not converge after 10 iterations of this process were removed from further analysis. In most cases these had severely multimodal delay distributions that were difficult to sample from and would have made further analysis difficult.

[1]

https://openalex.org/W2148534890

889554f9-7bfa-4fa1-823f-9479872c3230

The synthetic data were generated by fitting a GP with the MLP covariance [1]} to the Pol-II measurements of the gene TIPARP (ENSG00000163659), and numerically solving the mRNA level using Eq. (REF ) with the GP posterior mean as \(p(t)\) . The parameters used were: \(\Delta \in \lbrace 0, 10, 20, 30\rbrace \mathrm {\;min}\) , \(t_{1/2} =\log (2) / \alpha \in \lbrace 2, 4, 8, 16, 32, 64\rbrace \mathrm {\;min}\) , \(\beta _0 = 0.005\) , \(\beta = 0.03\) , \(m_0 = 0.008/\alpha \) . The parameter values were chosen empirically to get profiles that approximately fitted the actual mRNA observations while looking reasonable and informative across the entire range of parameter values.

[1]

https://openalex.org/W1746819321

21ef01bd-afcb-4d01-bbe6-713c2d34c1cc

Introduction. The statistics of individual speckle patterns created by coherent illumination of an optically random medium is well understood. It is widely accepted that as the real and imaginary parts of the scattered field are uncorrelated Gaussian random variables, the field amplitude follows a Rayleigh distribution and the light intensity follows a negative exponential distribution [1]}. The joint statistics of the scattered light is a more challenging problem. Various types of correlations in space, time or frequency may exist in the scattered fields, depending on the problem geometry and disorder strength [2]}, [3]}. A particularly relevant configuration involves imaging through a scattering slab, where the goal is to reconstruct the image from the transmitted scattered light. The strongest and most evident of the field correlations, the memory effect (ME) [4]}, [3]}, has proven to be useful in such a situation for example allowing to reconstruct the image from a simple autocorrelation calculation of the speckle pattern [6]}, [7]}, [8]}. Recently more subtle long-range mesoscopic correlations emerging in strongly scattering media [9]} have been used to retrieve the image of a hidden object [10]}. However, it is well known that correlations capture only a fraction of the total statistical dependence between random variables.

[1]

https://openalex.org/W137864573

e80038b9-398f-443f-8291-4ea7f8db99b9

As we go from surface to the center of a neutron star, at sufficiently high densities, the matter is expected to undergo a transition from hadronic matter, where the quarks are confined inside the hadrons, to a state of deconfined quarks. Finally, there are up, down and strange quarks in the quark matter. Other quarks have high masses and do not appear in this state. Glendenning has shown that a proper construction of the hadron-quark phase transition inside the neutron stars implies the coexistence of nucleonic matter and quark matter over a finite range of the pressure. Therefore, a mixed hadron-quark phase exists in the neutron star and its energy is lower than those of the quark matter and nucleonic matter [1]}. These show that we can consider a neutron star as composed of a hadronic matter layer, a mixed phase of quarks and hadrons and, in core, a quark matter. Recent Chandra observations also imply that the objects RX J185635-3754 and 3C58 could be neutron stars with the quark core [2]}.

[1]

https://openalex.org/W2073220558

b449e642-6b34-4481-b1da-163640581471

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[2]

https://openalex.org/W2061136215

9084b508-c8dd-4b1b-9823-310dfbcd30b7

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[3]

https://openalex.org/W1991212332

82867e4a-659e-4fa3-93ed-db9fe10962ca

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[4]

https://openalex.org/W2071489567

4efa1d94-9bfe-46b4-af1b-0feb5ffbfb89

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[6]

https://openalex.org/W2015172574

afbd7c2f-3073-478c-aac4-dfdd2be27dee

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[7]

https://openalex.org/W2044584848

1f859c7d-c1c5-481d-8ea8-fe7474b8d25e

In our calculations, the equation of state of hot nucleonic matter is determined using the lowest order constrained variational (LOCV) method as follows [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We adopt a trail wave function as \(\psi =F\phi ,\)

[8]

https://openalex.org/W2047803897

8a06c512-3ed4-4dc0-9c36-59d025ddb92b

The procedure to calculate the nucleonic matter has been fully discussed in the Refs. [1]}, [2]}.

[2]

https://openalex.org/W2061136215

d6090bd9-0de3-4102-819e-b5f226708452

We use the MIT bag model for the quark matter calculations. In this model, the energy density is the kinetic energy of quarks plus a bag constant (\({\cal B}\) ) which is interpreted as the difference between the energy densities of non interacting quarks and interacting ones [1]}, \({\cal E}_{tot} = {\cal E}_u + {\cal E}_d + {\cal E}_s + {\cal B},\)

[1]

https://openalex.org/W4242783256

c1bd2c03-3c09-4060-bb0b-07de5257d8fa

For the mixed phase, where the fraction of space occupied by quark matter smoothly increases from zero to unity, we have a mixture of hadrons, quarks and electrons. In the mixed phase, according the Gibss equilibrium condition, the temperatures, pressures and chemical potentials of the hadron phase (H) and quark phase (Q) are equal [1]}. Here, for each temperature we let the pressure to be an independent variable.

[1]

https://openalex.org/W2073220558

e52e8ebe-4b40-4d0e-b76f-3b6775bfb31a

To obtain \(\mu ^H_p\) and \(\mu ^H_n\) for the hadronic matter in mixed phase, we use the semiempirical mass formula [1]}, [2]}, [3]}, \(E=T(n,x)+V_0(n)+(1-2x)^2V_2(n),\)

[3]

https://openalex.org/W2057910081

fbed2da0-7d34-479e-8a0b-6eb673979bd6

The structure of neutron star is determined by numerically integrating the Tolman-Oppenheimer-Volkoff equation (TOV) [1]}, [2]}, [3]}, [4]}, \(\frac{dP}{dr}=-\frac{G[{\cal E}(r)+\frac{P(r)}{c^2}][m(r)+\frac{4\pi r^3 P(r)}{c^2}]}{r^2[1-\frac{2Gm(r)}{rc^2}]},\) \(\frac{dm}{dr}=4\pi r^2{\cal E}(r),\)

[1]

https://openalex.org/W2498126760

d3d1c884-baeb-4654-a279-469c1ade3f35

The structure of neutron star is determined by numerically integrating the Tolman-Oppenheimer-Volkoff equation (TOV) [1]}, [2]}, [3]}, [4]}, \(\frac{dP}{dr}=-\frac{G[{\cal E}(r)+\frac{P(r)}{c^2}][m(r)+\frac{4\pi r^3 P(r)}{c^2}]}{r^2[1-\frac{2Gm(r)}{rc^2}]},\) \(\frac{dm}{dr}=4\pi r^2{\cal E}(r),\)

[2]

https://openalex.org/W1649945517

592cb1e6-e0c7-4874-98e9-16718fbc388c

In our calculations for the structure of hot neutron star with the quark core, we use the following equations of state: (i) Below the density of \(0.05\ fm^3\) , we use the equation of state calculated by Baym [1]}. (ii) From the density of \(0.05\ fm^3\) up to the density where the mixed phase starts, we use the equation of state of pure hadron phase calculated in section REF . (iii) In the range of densities in which there is the mixed phase, we use the equation of state calculated in section REF . (iv) Beyond the density of end point of the mixed phase, we use the equation of state of pure quark phase calculated in section REF . All calculations are done for \({\cal B}=90\ MeV fm^{-3}\) at two different temperatures \(T=10\) and \(20\ MeV\) . Our results are as follows.

[1]

https://openalex.org/W4235382658

00f7f695-dfd2-4fcb-a629-7a10c89f4ba5

To address such problems in abstractive summarization, several works have been proposed to improve the factual consistency of summarization. As shown in Table REF , existing works can be roughly categorized into two classes: fact-input methods [1]}, [2]} which aim to encode the information of facts in the article, and post-edit methods [3]}, [4]} which seek to correct the factual errors after decoding. What's more, there are some integrated works which perform both improvements [5]}. These methods usually need to modify the architecture of the model, adding additional encoding modules or post-edit modules. <FIGURE>

[1]

https://openalex.org/W2963676814

92c1de2d-8518-4ad6-80b7-05448df12e5e

To address such problems in abstractive summarization, several works have been proposed to improve the factual consistency of summarization. As shown in Table REF , existing works can be roughly categorized into two classes: fact-input methods [1]}, [2]} which aim to encode the information of facts in the article, and post-edit methods [3]}, [4]} which seek to correct the factual errors after decoding. What's more, there are some integrated works which perform both improvements [5]}. These methods usually need to modify the architecture of the model, adding additional encoding modules or post-edit modules. <FIGURE>

[3]

https://openalex.org/W3102645206

d68fd688-4e94-4095-865e-c1a3cc9abc03

Contrastive learning on encoder (CoEnc) calculates the contrastive loss described in [1]}. CoEnc first encodes the article and summaries (ground truth and negative samples), then make the encoded representation of the article and the ground truth summary closer, and make that of the article and the factual inconsistency summary apart. As shown in Figure REF , the motivation of encoding both article and summary on the encoder is to catch and encode fact information. Given the article, the encoder can only distinguish the ground truth summary from the very similar negative summary by catching the common correct fact in the article-summary pair. It can be also explained from the view of data augmentation. Similar to the crops and rotations of images in SimCLR [2]}, we can regard the article and summary as two kinds of “data augmentation” on the fact. CO2Sum is designed to catch the fact behind the augmentation.

[2]

https://openalex.org/W3005680577

901c71ef-49de-4203-a5c0-715c670599ed

QAGS [1]}: QAGS generates questions about named entities and noun phrases in the predicted summary using a trained QG (Question Generation) model, then uses a QA (Question Answering) model to find answers to questions from the corresponding article. QAGS calculates token-level F1 similarity between QA results and asked entities or noun phrase in the summary as the final score. QuestEval [2]}: Compared to QAGS, QuestEval considers the situation of unanswerable questions. What's more, QuestEval does not calculate answer similarity, but scores the precision and recall apart, then QuestEval gives a weighted F1 score. Close Scheme Fact Triple [3]}: Fact Triple based metrics score the precision between summary extracted triple and article extracted triple. The triples \((Subject, Relation, Object)\) are extracted using Named Entity Recognition (NER) and Relation Extraction (RE) models. These triples are structured data of factual information and can be used to evaluate factual consistency. Open Scheme Fact Triple [3]}: Open Scheme Fact Triple is similar to the close ones, but the relationship in the fact triple is text span instead of classified relation label.

[3]

https://openalex.org/W2947681066

b37ab257-61fd-41c7-94e1-c0c181a92f44

We use factsummhttps://github.com/Huffon/factsumm [1]}, OpenIEhttps://github.com/philipperemy/stanford-openie-python [2]} and official provided codehttps://github.com/ThomasScialom/QuestEval [3]} to build evaluation system. On the use of trained models, we choose FLAIR [4]} for NER, LUKE [5]} for RE, T5 [6]} and Roberta [7]} for QA and QG. We calculate all Fact Triple metrics only on oracle sentences in article and summaries, since there is no need to calculate triple precision on those redundant sentences in the article.

[2]

https://openalex.org/W2251913848

e04672b2-2bf0-4284-b0f4-1134d2a2f3e8

Random: randomly pick and replace words in the ground truth summary to construct negative samples. NP: identify noun phrases in the summary and replace the words in the phrase. NER: perform Named Entity Recognition on the summary and construct negative samples with entity-level replacements. LFN: the proposed language model based construction method in CO2Sum. LFN (DN): similar to the improvement of Roberta [1]} over BERT [2]}, we perform dynamic (DN) negative sample construction during training.

[2]

https://openalex.org/W2896457183

4fff9dae-4fcf-45ad-9b5d-5875b98a6b50

In this section, we study the loss function in the CoDec. We compare the results of PM max-margin loss with the original loss (Vanilla) described in [1]}. Besides, we attempt another gated-weighting method (Gated) that dynamically calculates the weight of different positions. It uses a Linear Gate Unit [2]} to calculate the weights based on the hidden state of the decoder. The results are shown in Table REF . The gated method does not perform better than the vanilla, but position masked loss outperforms vanilla on all metrics. We assume that it is too difficult for a model to learn the different weights of positions. A simple mask can stabilize the training and performs better. <FIGURE><FIGURE>

[2]

https://openalex.org/W2964265128

aa6fa84c-696b-4729-8f7e-4099f59bc303

Most existing methods for improving fact consistency can be divided into fact-input-based methods and post-edit-based methods. Fact-input-based methods focus on enhancing the representation of facts in the source article or incorporating commonsense knowledge, which is useful to facilitate summarization systems understanding the facts for reducing consistent error. [1]} introduce FTSum to reduce consistent error by applying the encoder to incorporate the fact description. [2]} aim to incorporate entailment knowledge into the summarization model. Post-edit based method aims to apply a post-edit on the model-generated summaries for obtaining more factual-consistent summarization. [3]} propose a fact corrector, which corrects the factual error in the model-generated summary in an iterative and auto-regressive manner. [4]} propose a neural-based corrector module to address the factual inconsistent issue by identifying and correcting factual errors in generated summaries. [5]} explore to model the facts in the source article with knowledge graphs based on a neural network. [6]} study contrast candidate generation and selection to correct the extrinsic fact hallucinations in a post-edit manner. Comparing with the above works, we aim to improve factual consistency through contrastive learning without introducing extra parameters.

[1]

https://openalex.org/W2963676814

59cd90bd-1f37-4d9a-a694-382219db96c5

Most existing methods for improving fact consistency can be divided into fact-input-based methods and post-edit-based methods. Fact-input-based methods focus on enhancing the representation of facts in the source article or incorporating commonsense knowledge, which is useful to facilitate summarization systems understanding the facts for reducing consistent error. [1]} introduce FTSum to reduce consistent error by applying the encoder to incorporate the fact description. [2]} aim to incorporate entailment knowledge into the summarization model. Post-edit based method aims to apply a post-edit on the model-generated summaries for obtaining more factual-consistent summarization. [3]} propose a fact corrector, which corrects the factual error in the model-generated summary in an iterative and auto-regressive manner. [4]} propose a neural-based corrector module to address the factual inconsistent issue by identifying and correcting factual errors in generated summaries. [5]} explore to model the facts in the source article with knowledge graphs based on a neural network. [6]} study contrast candidate generation and selection to correct the extrinsic fact hallucinations in a post-edit manner. Comparing with the above works, we aim to improve factual consistency through contrastive learning without introducing extra parameters.

[3]

https://openalex.org/W3102489149

f114c42f-8206-4596-81f4-044a594d00a3

Most existing methods for improving fact consistency can be divided into fact-input-based methods and post-edit-based methods. Fact-input-based methods focus on enhancing the representation of facts in the source article or incorporating commonsense knowledge, which is useful to facilitate summarization systems understanding the facts for reducing consistent error. [1]} introduce FTSum to reduce consistent error by applying the encoder to incorporate the fact description. [2]} aim to incorporate entailment knowledge into the summarization model. Post-edit based method aims to apply a post-edit on the model-generated summaries for obtaining more factual-consistent summarization. [3]} propose a fact corrector, which corrects the factual error in the model-generated summary in an iterative and auto-regressive manner. [4]} propose a neural-based corrector module to address the factual inconsistent issue by identifying and correcting factual errors in generated summaries. [5]} explore to model the facts in the source article with knowledge graphs based on a neural network. [6]} study contrast candidate generation and selection to correct the extrinsic fact hallucinations in a post-edit manner. Comparing with the above works, we aim to improve factual consistency through contrastive learning without introducing extra parameters.

[4]

https://openalex.org/W3102645206

3459e464-0d64-4e71-b268-2ee671be3232

The orthogonal and unitary calculi allow for the systematic study of functors from either the category of real inner product spaces, or the category of complex inner product spaces, to the category of based topological spaces. The motivating examples are \(\mathrm {BO}(-) \colon V \longmapsto \mathrm {BO}(V)\) , where \(\mathrm {BO}(V)\) is the classifying space of the orthogonal group of \(V\) , and \(\mathrm {BU}(-)\colon W \longmapsto \mathrm {BU}(W)\) where \(\mathrm {BU}(W)\) is the classifying space of the unitary group of \(W\) . The foundations of orthogonal calculus were originally developed by Weiss in [1]}, and later converted to a model category theoretic framework by Barnes and Oman in [2]}. From the unitary calculus perspective, it has long been known to the experts, with the foundations and model category framework developed by the author in [3]}.

[1]

https://openalex.org/W4239559895

7bf5ee4a-2b35-43c9-b8c7-bce6275a749c

The orthogonal and unitary calculi allow for the systematic study of functors from either the category of real inner product spaces, or the category of complex inner product spaces, to the category of based topological spaces. The motivating examples are \(\mathrm {BO}(-) \colon V \longmapsto \mathrm {BO}(V)\) , where \(\mathrm {BO}(V)\) is the classifying space of the orthogonal group of \(V\) , and \(\mathrm {BU}(-)\colon W \longmapsto \mathrm {BU}(W)\) where \(\mathrm {BU}(W)\) is the classifying space of the unitary group of \(W\) . The foundations of orthogonal calculus were originally developed by Weiss in [1]}, and later converted to a model category theoretic framework by Barnes and Oman in [2]}. From the unitary calculus perspective, it has long been known to the experts, with the foundations and model category framework developed by the author in [3]}.

[2]

https://openalex.org/W2034710066

d84fb767-a4bb-4c4d-975b-1e55556ab617

We start with the categories of spectra in Section , and use the change of group functors of Mandell and May [1]}, to construct Quillen adjunctions between spectra with an action of \(\mathrm {O}(n)\) , \(\mathrm {U}(n)\) and \(\mathrm {O}(2n)\) respectively. We utilise the Quillen equivalence between orthogonal and unitary spectra of [2]} to show that these change of group functors interact in a homotopically meaningful way with the change of model functor induced by realification.

[1]

https://openalex.org/W2075488415

84189aac-5c29-4c97-9c75-bec1acac3733

In Section we move to comparing the intermediate categories. These are categories \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {O}\) and \(\mathrm {U}(n)\mathcal {E}_n^\mathbf {U}\) constructed by Barnes and Oman [1]}, and the author [2]}, which act as an intermediate in the zig-zag of Quillen equivalences for orthogonal and unitary calculus respectively. For this, we introduce two new intermediate categories, \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {U}\) and \(\mathrm {U}(n)\mathcal {E}_{2n}^\mathbf {O}\) between the standard intermediate categories. These are the standard intermediate categories with restricted group actions through the subgroup inclusions \(\mathrm {O}(n) \hookrightarrow \mathrm {U}(n)\) and \(\mathrm {U}(n) \hookrightarrow \mathrm {O}(2n)\) . We exhibit Quillen equivalences between these intermediate categories and the standard intermediate categories, completing the picture using change of group functors from [3]}. The resulting diagram of intermediate categories is as follows, \(@C=4em{\mathrm {O}(n)\mathcal {E}_n^\mathbf {O} @<-1ex>[r]_{r^*}^\sim & \mathrm {O}(n)\mathcal {E}_n^\mathbf {U} @<1ex>[r]^{\mathrm {U}(n)_+\wedge _{\mathrm {O}(n)}(-)}@<-1ex>[l]_{r_!} & \mathrm {U}(n)\mathcal {E}_n^\mathbf {U} @<1ex>[l]^{\iota ^*}@<-1ex>[r]_{c^*} & \mathrm {U}(n)\mathcal {E}_{2n}^\mathbf {O} @<1ex>[r]^{\mathrm {O}(2n)_+\wedge _{\mathrm {U}(n)}(-)} @<-1ex>[l]_{c_!}^\sim & \mathrm {O}(2n)\mathcal {E}_{2n}^\mathbf {O}. @<1ex>[l]^{\kappa ^*}}\)

[1]

https://openalex.org/W2034710066

c3295b9d-b52d-479c-a9bb-09debbaf0751

In Section we move to comparing the intermediate categories. These are categories \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {O}\) and \(\mathrm {U}(n)\mathcal {E}_n^\mathbf {U}\) constructed by Barnes and Oman [1]}, and the author [2]}, which act as an intermediate in the zig-zag of Quillen equivalences for orthogonal and unitary calculus respectively. For this, we introduce two new intermediate categories, \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {U}\) and \(\mathrm {U}(n)\mathcal {E}_{2n}^\mathbf {O}\) between the standard intermediate categories. These are the standard intermediate categories with restricted group actions through the subgroup inclusions \(\mathrm {O}(n) \hookrightarrow \mathrm {U}(n)\) and \(\mathrm {U}(n) \hookrightarrow \mathrm {O}(2n)\) . We exhibit Quillen equivalences between these intermediate categories and the standard intermediate categories, completing the picture using change of group functors from [3]}. The resulting diagram of intermediate categories is as follows, \(@C=4em{\mathrm {O}(n)\mathcal {E}_n^\mathbf {O} @<-1ex>[r]_{r^*}^\sim & \mathrm {O}(n)\mathcal {E}_n^\mathbf {U} @<1ex>[r]^{\mathrm {U}(n)_+\wedge _{\mathrm {O}(n)}(-)}@<-1ex>[l]_{r_!} & \mathrm {U}(n)\mathcal {E}_n^\mathbf {U} @<1ex>[l]^{\iota ^*}@<-1ex>[r]_{c^*} & \mathrm {U}(n)\mathcal {E}_{2n}^\mathbf {O} @<1ex>[r]^{\mathrm {O}(2n)_+\wedge _{\mathrm {U}(n)}(-)} @<-1ex>[l]_{c_!}^\sim & \mathrm {O}(2n)\mathcal {E}_{2n}^\mathbf {O}. @<1ex>[l]^{\kappa ^*}}\)

[3]

https://openalex.org/W2075488415

d055ae69-d195-47fe-93d4-e1e9b8511910

The category \(\mathcal {E}_0^\mathbf {O}\) is category of input functors for orthogonal calculus as studied by Weiss and Barnes and Oman [1]}, [2]}. Moreover \(\mathcal {E}_0^\mathbf {U}\) is the category of input functors for unitary calculus, studied by the author in [3]}. These input categories are categories of diagram spaces as in [4]} hence they can be equipped with a projective model structure.

[1]

https://openalex.org/W4239559895

fc1ecf49-4b1e-4e84-a134-0b1334658e79

The category \(\mathcal {E}_0^\mathbf {O}\) is category of input functors for orthogonal calculus as studied by Weiss and Barnes and Oman [1]}, [2]}. Moreover \(\mathcal {E}_0^\mathbf {U}\) is the category of input functors for unitary calculus, studied by the author in [3]}. These input categories are categories of diagram spaces as in [4]} hence they can be equipped with a projective model structure.

[2]

https://openalex.org/W2034710066

8c9c4505-b813-4cce-8fee-57cb0b866676

The category \(\mathcal {E}_0^\mathbf {O}\) is category of input functors for orthogonal calculus as studied by Weiss and Barnes and Oman [1]}, [2]}. Moreover \(\mathcal {E}_0^\mathbf {U}\) is the category of input functors for unitary calculus, studied by the author in [3]}. These input categories are categories of diagram spaces as in [4]} hence they can be equipped with a projective model structure.

[4]

https://openalex.org/W2086997195

40147305-f885-4934-908d-5d29019b0e32

Arguably the most important class of functors in orthogonal and unitary calculi are the \(n\) -polynomial functors, and in particular the \(n\) -th polynomial approximation functor. Here we give a short overview of these functors, for full details on these functors see [1]}, [2]}, [3]}.

[1]

https://openalex.org/W4239559895

f12275ec-379c-41b0-ac5b-b0da9c85b7cc

Arguably the most important class of functors in orthogonal and unitary calculi are the \(n\) -polynomial functors, and in particular the \(n\) -th polynomial approximation functor. Here we give a short overview of these functors, for full details on these functors see [1]}, [2]}, [3]}.

[2]

https://openalex.org/W2034710066

fb817270-e1c1-4c09-add2-fb9d1d65073b

Since an \(n\) -polynomial functor is \((n+1)\) -polynomial, see [1]}, these polynomial approximation functors assemble into a Taylor tower approximating a given input functor. Moreover there is a model structure on \(\mathcal {E}_0\) which captures the homotopy theory of \(n\) -polynomial functors.

[1]

https://openalex.org/W4239559895

cdbca8f4-1140-4d41-a723-9ae62128df76

Proposition 2.5 ([1]}, [2]}) There is a cellular proper topological model structure on \(\mathcal {E}_0\) where a map \(f\colon E \rightarrow F\) is a weak equivalence if \(T_nf\colon T_nE \rightarrow T_nF\) is a levelwise weak equivalence, the cofibrations are the cofibrations of the projective model structure and the fibrations are levelwise fibrations such that \({E [r]^f [d]_{\eta _E} & F [d]^{\eta _F} \\T_nE [r]_{T_nf} & T_n F}\)

[1]

https://openalex.org/W2034710066

8b53fb8c-ee82-4b7c-9099-462995e8f439

Proposition 2.7 ([1]}, [2]}) There is a topological model structure on \(\mathcal {E}_0\) where the weak equivalences are those maps \(f\) such that \(D_nf\) is a weak equivalence in \(\mathcal {E}_0\) , the fibrations are the fibrations of the \(n\) -polynomial model structure and the cofibrations are those maps with the left lifting property with respect to the acyclic fibrations. The fibrant objects are \(n\) -polynomial and the cofibrant-fibrant objects are the projectively cofibrant \(n\) -homogeneous functors.

[1]

https://openalex.org/W2034710066

519d46f4-b116-4f80-b76b-a3ae1c3b1904

In [1]}, Weiss constructs a zig-zag of equivalences between the category of \(n\) -homogeneous functors (up to homotopy) and the homotopy category of spectra with an action of \(\mathrm {O}(n)\) . In [2]}, Barnes and Oman put this zig-zag into a model category theoretic framework via a zig-zag of Quillen equivalences between the \(n\) -homogeneous model structure on \(\mathcal {E}_0^\mathbf {O}\) , and spectra with an action of \(\mathrm {O}(n)\) . This zig-zag moves through an intermediate category, denote \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {O}\) . In [3]}, the author constructs a similar zig-zag of Quillen equivalences between the unitary \(n\) -homogeneous model structure and spectra with an action of \(\mathrm {U}(n)\) . We give an overview of the construction of these intermediate categories and how they relate to spectra and the \(n\) -homogenous model structure.

[1]

https://openalex.org/W4239559895

d36d32a8-5b37-417d-b405-de0ddf3fd78c

In [1]}, Weiss constructs a zig-zag of equivalences between the category of \(n\) -homogeneous functors (up to homotopy) and the homotopy category of spectra with an action of \(\mathrm {O}(n)\) . In [2]}, Barnes and Oman put this zig-zag into a model category theoretic framework via a zig-zag of Quillen equivalences between the \(n\) -homogeneous model structure on \(\mathcal {E}_0^\mathbf {O}\) , and spectra with an action of \(\mathrm {O}(n)\) . This zig-zag moves through an intermediate category, denote \(\mathrm {O}(n)\mathcal {E}_n^\mathbf {O}\) . In [3]}, the author constructs a similar zig-zag of Quillen equivalences between the unitary \(n\) -homogeneous model structure and spectra with an action of \(\mathrm {U}(n)\) . We give an overview of the construction of these intermediate categories and how they relate to spectra and the \(n\) -homogenous model structure.

[2]

https://openalex.org/W2034710066

18440549-2077-4c01-af6d-1eb0976a5c8a

Let \(n\mathbb {S}\) be the functor given by \(V \longmapsto S^{nV}\) where \(nV := \mathbb {F}^n \otimes _\mathbb {F}V\) . By [1]} and [2]} the intermediate categories are equivalent to a category of \(n\mathbb {S}\) -modules and hence come equipped with an \(n\) -stable model structure similar to the stable model structure on spectra. The weak equivalences of the \(n\) -stable model structure are given by \(n\pi _*\) -isomorphisms. Theses are defined via the structure maps of objects in \(\mathrm {Aut}(n)\mathcal {E}_n\) , and as such have slightly different forms depending on whether one is in the orthogonal or unitary setting.

[1]

https://openalex.org/W2034710066

76319983-df13-43b2-8825-7593fa9945ba

Proposition 2.14 ([1]}, [2]}) There is a cofibrantly generated, proper, topological model structure on the category \(\mathrm {Aut}(n)\mathcal {E}_n\) , where the weak equivalences are the \(n\pi _*\) -isomorphisms, the cofibrations are those maps with the left lifting property with respect to all levelwise acyclic fibrations and the fibrations are those levelwise fibrations \(f\colon X \rightarrow Y\) such that the diagram \({X(V) [r] [d] & \Omega ^{nW}X(V \oplus W) [d] \\Y(V) [r] & \Omega ^{nW}Y(V \oplus W).}\)

[1]

https://openalex.org/W2034710066

0e8071a3-282f-4fc6-803d-90229e47e5df

The orthogonal case is similar, full details may be found in [1]}.

[1]

https://openalex.org/W2034710066

1defc0aa-19c7-4b56-b6a6-fcc317b64634

Proposition 2.16 ([1]}) There is a Quillen equivalence \({(\beta _n)_!:\mathrm {O}(n)\mathcal {E}_n^\mathbf {O} @<0.7ex>[r] &@<0.7ex>[l] \mathsf {Sp}^\mathbf {O}[\mathrm {O}(n)]:(\beta _n)^*}\)

[1]

https://openalex.org/W2034710066

37583206-78d3-4cf4-b168-7414de42c1be

Combining this adjunction with a change of group action from [1]} provides an adjunction \({\operatorname{\mathrm {res}}_0^n/\mathrm {Aut}(n):\mathrm {Aut}(n)\mathcal {E}_n @<0.7ex>[r] &@<0.7ex>[l] \mathcal {E}_0:\operatorname{\mathrm {ind}}_0^n \varepsilon ^*}.\)

[1]

https://openalex.org/W2075488415

27c19ee1-26ce-4a58-8dbc-6f8427360208

Proposition 2.18 ([1]}, [2]}) The adjoint pair \({\operatorname{\mathrm {res}}_0^n/\mathrm {Aut}(n):\mathrm {Aut}(n)\mathcal {E}_n @<0.7ex>[r] &@<0.7ex>[l] n\operatorname{--homog--}\mathcal {E}_0:\operatorname{\mathrm {ind}}_0^n\varepsilon ^*}\)

[1]

https://openalex.org/W2034710066

eff0da6b-b918-45d9-bbd2-dbead0678a91

Proposition 2.19 ([1]},[2]}) Let \(F \in \mathcal {E}_0\) be \(n\) -homogeneous for some \(n >0\) . Then \(F\) is levelwise weakly equivalent to the functor defined as \(U \longmapsto \Omega ^\infty [(S^{nU} \wedge \Psi _F^n)_{h\mathrm {Aut}(n)}].\)

[1]

https://openalex.org/W4239559895

5a7b973c-95d3-4736-a1fc-063bccd2808e

When two functors agree to a given order, their Taylor tower agree to a prescribed level. The first result in that direction is the unitary analogue of [1]}.

[1]

https://openalex.org/W4239559895

5c3089fd-1790-4c03-af63-b68a736fa192

Lemma 2.21 ([1]},[2]}) let \(p \colon G \rightarrow F\) be a map in \(\mathcal {E}_0\) . Suppose that there is \(b \in \mathbb {Z}\) such that \(p_U\colon G(U) \rightarrow F(U)\) is \(((n+1)\dim _\mathbb {R}(U) - b)\) -connected for all \(U \in J_0\) with \(\dim _\mathbb {F}(U) \ge \rho \) . Then \(\tau _n(p)_U \colon \tau _n (G(U)) \rightarrow \tau _n(F(U))\)

[1]

https://openalex.org/W4239559895

725a69a7-6c3c-497f-b696-827d786e5479

Lemma 2.22 ([1]},[2]}) If \(p \colon F \rightarrow G\) is an order \(n\) agreement, then \(T_k F \rightarrow T_k G\) is a levelwise weak equivalence for \(k \le n\) .

[1]

https://openalex.org/W4239559895

50009f15-6c5d-496e-b338-3e37f3389027

Let \(F\) be an \(n\) -homogeneous orthogonal functor. Then by the characterisation, Proposition REF , \(F\) is levelwise weakly equivalent to the functor \(V \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}V} \wedge \Psi _F^n)_{h\mathrm {O}(n)}]\) where \(\Psi _F^n\) is an orthogonal spectrum with an \(O(n)\) -action. It follows that pre-realification of \(F\) is levelwise weakly equivalent to the functor \(W \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}} \wedge \Psi ^n_F)_{h\mathrm {O}(n)}].\) Using the derived change of group functor, we construct an orthogonal spectrum with an action of \(\mathrm {U}(n)\) , \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)}^{L} \Psi _F^n := \mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} (E\mathrm {O}(n)_+ \wedge \Psi _F^n).\) By the classification of \(n\) -homogeneous unitary functors, Proposition REF , there is an \(n\) -homogeneous functor \(F^{\prime }\) associated to the above spectrum, given by \(W \longmapsto \Omega ^\infty [(S^{n \otimes _W̏} \wedge (\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))_{h\mathrm {U}(n)}].\) By [1]}, \(F^{\prime }(W)\) is isomorphic to \(\Omega ^\infty [\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))_{h(\mathrm {U}(n)}].\) The \(\mathrm {U}(n)\) -action on \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))\) is free (\((E\mathrm {O}(n)_+\) is a free \(\mathrm {O}(n)\) -space), hence taking homotopy orbits equates to taking strict orbits. Hence there is an isomorphism \(F^{\prime }(W) \cong \Omega ^\infty [(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))))/\mathrm {U}(n)].\) The strict \(\mathrm {U}(n)\) -orbits of the spectrum \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))\) are isomorphic to the \(\mathrm {O}(n)\) -orbits of the spectrum, \(\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))\) , hence \(F^{\prime }(W)\) is isomorphic to \(\Omega ^\infty [(\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))/\mathrm {O}(n)].\) This last is precisely \(\Omega ^\infty [(\iota ^*S^{n \otimes _W̏} \wedge \Psi _F^n)_{h\mathrm {O}(n)}]\) as homotopy orbits is the left derived functor of strict orbits and smashing with \(E\mathrm {O}(n)_+\) is a cofibrant replacement in the projective model structure. Since the action of \(\mathrm {O}(n)\) on \(\iota ^*S^{n \otimes _W̏}\) is equivalent to the \(\mathrm {O}(n)\) action on \(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}}\) and the one-point compactification are isomorphic, the above infinite loop space is isomorphic to \(\Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}} \wedge \Psi _F^n)_{h\mathrm {O}(n)}].\) By the characterisation of \(n\) -homogeneous orthogonal functors, we see that this is levelwise weakly equivalent to \(F(W_\mathbb {R}) = (r^*F)(W).\)

[1]

https://openalex.org/W2075488415

8b54c81b-7689-4d80-a5fa-142307906e26

We argue by induction on the polynomial degree. The case \(n=0\) follows by definition. Assume the map \(r^*T_{n-1}^\mathbf {O} F \rightarrow T_{n-1}^\mathbf {U}(r^*T_{n-1}^\mathbf {O} F)\) is a levelwise weak equivalence. There is a homotopy fibre sequence \(T_n^\mathbf {O} F \longrightarrow T_{n-1}^\mathbf {O} F \longrightarrow R_n^\mathbf {O} F\) where \(R_n^\mathbf {O} F\) is \(n\) -homogeneous, since \(F\) satisfies the conditions of [1]}. Lemma REF implies that \(r^* R_n^\mathbf {O} F\) is \(n\) -homogeneous in \(\mathcal {E}_0^\mathbf {U}\) , and in particular \(n\) -polynomial. As homotopy fibres of maps between \(n\) -polynomial objects are \(n\) -polynomial, the homotopy fibre of the map \(r^*T_{n-1}^\mathbf {O} F \rightarrow r^* R_n^\mathbf {O} F\) is \(n\) -polynomial. Computation of homotopy fibres is levelwise, hence the homotopy fibre in question is \(r^* T_n^\mathbf {O} F\) , and it follows that \(r^* T_n^\mathbf {O} F \longrightarrow T_n^\mathbf {U}(r^* T_n^\mathbf {O} F)\) is a levelwise weak equivalence.

[1]

https://openalex.org/W4239559895

e4c19355-c6f1-4ebf-a76b-accaa5c9cb87

The acyclic fibrations in the \(n\) –polynomial model structure are levelwise, hence both terms in the composite, and resultingly their composite, preserves these. It suffices by [1]} to show that the composite preserves fibrations between fibrant objects, which are the levelwise fibrations by [2]}. It hence suffices to show that \(F\) preserves fibrant objects. Let \(F\) be a \(n\) –polynomial orthogonal functor, then \(\mathrm {red}(F)\) is also \(n\) –polynomial, and reduced. An application of Theorem REF implies the result.

[2]

https://openalex.org/W1583122470

2825ac31-7e00-48d1-97d5-5976dee8b1e0

The notion of agreement plays a central role in the theory or orthogonal and unitary calculus, for example it is crucial to the proof that the \(n\) -th polynomial approximation in \(n\) -polynomial, see [1]}. The pre-realification and pre-complexification functors behave well with respect to functors which agree to a certain order.

[1]

https://openalex.org/W4239559895

6158a9f6-e00d-41e9-bd26-082f31840430

We have constructed a Quillen adjunction between the orthogonal and unitary \(n\) -homogeneous model structures. To give a complete comparison of the theories we must address the comparisons between the other two categories in the zig-zag of Quillen equivalences of Barnes and Oman [1]} and the author [2]}. We start by addressing the relationship between the categories of spectra. For this, we recall the definitions and model structures involved.

[1]

https://openalex.org/W2034710066

0a05370c-d6e6-45a6-a674-ddf8906dda09

compare [1]}.

[1]

https://openalex.org/W2075488415

e1946ea6-6ab7-40a2-8aef-b31c07426919

These categories also come with projective and stable model structures constructed analogously to those of Proposition REF . These new intermediate categories will now act as intermediate categories between the standard intermediate categories of orthogonal and unitary calculus. Further, the new intermediate categories equipped with their \(n\) -stable model structure are Quillen equivalent to spectra with an appropriate group action. The proofs of the following two results follow similarly to [1]} and [2]}.

[1]

https://openalex.org/W2034710066

6a555566-74ef-41dd-b36b-c0538abb7878

where the isomorphism follows from [1]}.

[1]

https://openalex.org/W2075488415

203637de-ef05-49dc-b95c-f4a04a88560c

By the zig-zag of Quillen equivalences, [1]} the composite \({L}\operatorname{\mathrm {res}}_0^n/\mathrm {O}(n) \circ {R}(\beta _n)^*\) applied to an orthogonal spectrum \(\Theta \) with an action of \(\mathrm {O}(n)\) , is levelwise weakly equivalent to the functor \(F\) , defined by the formula \(V \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes V} \wedge \Theta )_{h\mathrm {O}(n)}].\) This functor is \(n\) –homogeneous, hence also reduced, so \({R}r^*\mathrm {red}(F)\) is levelwise weakly equivalent to \(\mathbb {R}r^*F\) . The zig-zag of Quillen equivalences from unitary calculus, [2]}, together with inflating \(\Theta \) to a spectrum with an action of \(\mathrm {U}(n)\) gives a similar characterisation in terms of an \(n\) -homogeneous functor. The result then follows by Proposition REF .

[1]

https://openalex.org/W2034710066

61ccc52d-9145-40e1-bd57-48a46af10baa

Let \(F\) be an \(n\) -homogeneous orthogonal functor. Then by the characterisation, Proposition REF , \(F\) is levelwise weakly equivalent to the functor \(V \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}V} \wedge \Psi _F^n)_{h\mathrm {O}(n)}]\) where \(\Psi _F^n\) is an orthogonal spectrum with an \(O(n)\) -action. It follows that pre-realification of \(F\) is levelwise weakly equivalent to the functor \(W \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}} \wedge \Psi ^n_F)_{h\mathrm {O}(n)}].\) Using the derived change of group functor, we construct an orthogonal spectrum with an action of \(\mathrm {U}(n)\) , \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)}^{L} \Psi _F^n := \mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} (E\mathrm {O}(n)_+ \wedge \Psi _F^n).\) By the classification of \(n\) -homogeneous unitary functors, Proposition REF , there is an \(n\) -homogeneous functor \(F^{\prime }\) associated to the above spectrum, given by \(W \longmapsto \Omega ^\infty [(S^{n \otimes _W̏} \wedge (\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))_{h\mathrm {U}(n)}].\) By [1]}, \(F^{\prime }(W)\) is isomorphic to \(\Omega ^\infty [\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))_{h(\mathrm {U}(n)}].\) The \(\mathrm {U}(n)\) -action on \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))\) is free (\((E\mathrm {O}(n)_+\) is a free \(\mathrm {O}(n)\) -space), hence taking homotopy orbits equates to taking strict orbits. Hence there is an isomorphism \(F^{\prime }(W) \cong \Omega ^\infty [(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))))/\mathrm {U}(n)].\) The strict \(\mathrm {U}(n)\) -orbits of the spectrum \(\mathrm {U}(n)_+ \wedge _{\mathrm {O}(n)} ((\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))\) are isomorphic to the \(\mathrm {O}(n)\) -orbits of the spectrum, \(\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n))\) , hence \(F^{\prime }(W)\) is isomorphic to \(\Omega ^\infty [(\iota ^*S^{n \otimes _W̏} \wedge (E\mathrm {O}(n)_+ \wedge \Psi _F^n)))/\mathrm {O}(n)].\) This last is precisely \(\Omega ^\infty [(\iota ^*S^{n \otimes _W̏} \wedge \Psi _F^n)_{h\mathrm {O}(n)}]\) as homotopy orbits is the left derived functor of strict orbits and smashing with \(E\mathrm {O}(n)_+\) is a cofibrant replacement in the projective model structure. Since the action of \(\mathrm {O}(n)\) on \(\iota ^*S^{n \otimes _W̏}\) is equivalent to the \(\mathrm {O}(n)\) action on \(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}}\) and the one-point compactification are isomorphic, the above infinite loop space is isomorphic to \(\Omega ^\infty [(S^{\mathbb {R}^n \otimes _\mathbb {R}W_\mathbb {R}} \wedge \Psi _F^n)_{h\mathrm {O}(n)}].\) By the characterisation of \(n\) -homogeneous orthogonal functors, we see that this is levelwise weakly equivalent to \(F(W_\mathbb {R}) = (r^*F)(W).\)

[1]

https://openalex.org/W2075488415

ec524d83-4cff-40c6-9118-616eb4472427

We argue by induction on the polynomial degree. The case \(n=0\) follows by definition. Assume the map \(r^*T_{n-1}^\mathbf {O} F \rightarrow T_{n-1}^\mathbf {U}(r^*T_{n-1}^\mathbf {O} F)\) is a levelwise weak equivalence. There is a homotopy fibre sequence \(T_n^\mathbf {O} F \longrightarrow T_{n-1}^\mathbf {O} F \longrightarrow R_n^\mathbf {O} F\) where \(R_n^\mathbf {O} F\) is \(n\) -homogeneous, since \(F\) satisfies the conditions of [1]}. Lemma REF implies that \(r^* R_n^\mathbf {O} F\) is \(n\) -homogeneous in \(\mathcal {E}_0^\mathbf {U}\) , and in particular \(n\) -polynomial. As homotopy fibres of maps between \(n\) -polynomial objects are \(n\) -polynomial, the homotopy fibre of the map \(r^*T_{n-1}^\mathbf {O} F \rightarrow r^* R_n^\mathbf {O} F\) is \(n\) -polynomial. Computation of homotopy fibres is levelwise, hence the homotopy fibre in question is \(r^* T_n^\mathbf {O} F\) , and it follows that \(r^* T_n^\mathbf {O} F \longrightarrow T_n^\mathbf {U}(r^* T_n^\mathbf {O} F)\) is a levelwise weak equivalence.

[1]

https://openalex.org/W4239559895

873bc9ce-fc96-423e-bacb-88fde79e3dfd

The acyclic fibrations in the \(n\) –polynomial model structure are levelwise, hence both terms in the composite, and resultingly their composite, preserves these. It suffices by [1]} to show that the composite preserves fibrations between fibrant objects, which are the levelwise fibrations by [2]}. It hence suffices to show that \(F\) preserves fibrant objects. Let \(F\) be a \(n\) –polynomial orthogonal functor, then \(\mathrm {red}(F)\) is also \(n\) –polynomial, and reduced. An application of Theorem REF implies the result.

[2]

https://openalex.org/W1583122470

20d38e3a-d77c-47df-b074-6fce7b7fadec

The notion of agreement plays a central role in the theory or orthogonal and unitary calculus, for example it is crucial to the proof that the \(n\) -th polynomial approximation in \(n\) -polynomial, see [1]}. The pre-realification and pre-complexification functors behave well with respect to functors which agree to a certain order.

[1]

https://openalex.org/W4239559895

4cf27916-a6d0-4336-a606-32bfcf030c10

We have constructed a Quillen adjunction between the orthogonal and unitary \(n\) -homogeneous model structures. To give a complete comparison of the theories we must address the comparisons between the other two categories in the zig-zag of Quillen equivalences of Barnes and Oman [1]} and the author [2]}. We start by addressing the relationship between the categories of spectra. For this, we recall the definitions and model structures involved.

[1]

https://openalex.org/W2034710066

db9b6bbc-117d-4939-9a14-ab149b71437b

compare [1]}.

[1]

https://openalex.org/W2075488415

769e3527-e453-4f89-98a8-61ec30911928

These categories also come with projective and stable model structures constructed analogously to those of Proposition REF . These new intermediate categories will now act as intermediate categories between the standard intermediate categories of orthogonal and unitary calculus. Further, the new intermediate categories equipped with their \(n\) -stable model structure are Quillen equivalent to spectra with an appropriate group action. The proofs of the following two results follow similarly to [1]} and [2]}.

[1]

https://openalex.org/W2034710066

2181f64d-7200-4ec6-91d7-055dfe27a120

where the isomorphism follows from [1]}.

[1]

https://openalex.org/W2075488415

b0855016-27d8-4693-ba77-50723aa9e165

By the zig-zag of Quillen equivalences, [1]} the composite \({L}\operatorname{\mathrm {res}}_0^n/\mathrm {O}(n) \circ {R}(\beta _n)^*\) applied to an orthogonal spectrum \(\Theta \) with an action of \(\mathrm {O}(n)\) , is levelwise weakly equivalent to the functor \(F\) , defined by the formula \(V \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes V} \wedge \Theta )_{h\mathrm {O}(n)}].\) This functor is \(n\) –homogeneous, hence also reduced, so \({R}r^*\mathrm {red}(F)\) is levelwise weakly equivalent to \(\mathbb {R}r^*F\) . The zig-zag of Quillen equivalences from unitary calculus, [2]}, together with inflating \(\Theta \) to a spectrum with an action of \(\mathrm {U}(n)\) gives a similar characterisation in terms of an \(n\) -homogeneous functor. The result then follows by Proposition REF .

[1]

https://openalex.org/W2034710066

8be61fa0-2ba6-462c-9e82-0c3b83709155

These categories also come with projective and stable model structures constructed analogously to those of Proposition REF . These new intermediate categories will now act as intermediate categories between the standard intermediate categories of orthogonal and unitary calculus. Further, the new intermediate categories equipped with their \(n\) -stable model structure are Quillen equivalent to spectra with an appropriate group action. The proofs of the following two results follow similarly to [1]} and [2]}.

[1]

https://openalex.org/W2034710066

d9c60347-3403-4dab-a8c1-2d470b426e3e

where the isomorphism follows from [1]}.

[1]

https://openalex.org/W2075488415

729488d8-56b5-4a8a-bca0-07bd7c9f778b

By the zig-zag of Quillen equivalences, [1]} the composite \({L}\operatorname{\mathrm {res}}_0^n/\mathrm {O}(n) \circ {R}(\beta _n)^*\) applied to an orthogonal spectrum \(\Theta \) with an action of \(\mathrm {O}(n)\) , is levelwise weakly equivalent to the functor \(F\) , defined by the formula \(V \longmapsto \Omega ^\infty [(S^{\mathbb {R}^n \otimes V} \wedge \Theta )_{h\mathrm {O}(n)}].\) This functor is \(n\) –homogeneous, hence also reduced, so \({R}r^*\mathrm {red}(F)\) is levelwise weakly equivalent to \(\mathbb {R}r^*F\) . The zig-zag of Quillen equivalences from unitary calculus, [2]}, together with inflating \(\Theta \) to a spectrum with an action of \(\mathrm {U}(n)\) gives a similar characterisation in terms of an \(n\) -homogeneous functor. The result then follows by Proposition REF .

[1]

https://openalex.org/W2034710066

f032747c-5eac-418b-9556-3006178ab0f2

Advances in the ability to successfully train very deep neural networks have been key to improving performance in image recognition, language modeling, and many other domains [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. Graph Neural Networks (GNNs) are a family of deep networks that operate on graph structured data by iteratively passing learned messages over the graph's structure [7]}, [8]}, [9]}, [10]}. While GNNs are very effective in a wide variety of tasks [11]}, [12]}, [13]}, deep GNNs, which perform more than 5-10 message passing steps, have not typically yielded better performance than shallower GNNs [14]}, [15]}, [16]}. While in principle deep GNNs should have greater expressivity and ability to capture complex functions, it has been proposed that in practice “oversmoothing” [17]} and “bottleneck effects” [18]} limit the potential benefits of deep GNNs. The purpose of this work is to reap the benefits of deep GNNs while avoiding such limitations.

[18]

https://openalex.org/W3034190530

20440e8e-e71f-464e-96ee-7c0ffbcb379d

Oversmoothing is a proposed phenomenon where a GNN's latent node representations become increasing similar over successive steps of message passing [1]}. Once these representations are oversmoothed, adding further steps does not add expressive capacity, and so performance does not improve. Bottleneck effects are thought to limit the ability of a deep GNN to communicate information over long ranges, because as the number of steps increases and causes the receptive fields to grow, the intermediate nodes must propagate information between larger and larger sets of peripheral nodes, limiting the ability to precisely transmit individual pieces of information [2]}. We speculate that these are reasons why previous strong GNN results on the molecular property benchmarks we study here, Open Catalyst 2020 (OC20) [3]} and QM9 [4]}, report depths of only 4 [5]} and 8 [6]}, even with large hyperparameter searches.

[2]

https://openalex.org/W3034190530

fc5a2298-6a8a-4f4c-a8e7-56ba197e71e6

Scaling Graph Neural Networks with Depth. Recent work has aimed to understand why it is challenging to realise the benefits of training very deep GNNs [1]}. A key contribution has been the analysis of “oversmoothing” which describes how all node features become almost identical in GCNs after a few layers. Since first being noted in [2]} oversmoothing has been studied extensively and regularisation techniques have been suggested to overcome it [3]}, [4]}, [5]}, [6]}, [7]}, but these studies have not shown consistent improvements as depth increases. Another insight has been the analysis of the bottleneck effect in [8]} which may be alleviated by using a fully connected graph. Finally, deep GNNs have been trained using techniques developed for CNNs [9]}, [10]}, but with no conclusive improvement from depth.

[8]

https://openalex.org/W3034190530

f43d9699-4709-425d-96a2-9e74b27e6321

An alternative approach to embedding symmetries is to design a rotation equivariant neural network that predicts quantities such as forces directly [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. However, the benefits of such models are typically seen on smaller data (e.g. [7]}), and such approaches have yet to achieve SOTA performance on large quantum chemistry datasets. Not all competitive models embed rotation symmetries—notably ForceNet [9]} does not enforce rotation equivariance.

[6]

https://openalex.org/W2970663744

f78f802e-9c86-44c6-8045-ead0b4c44fef

Processor. The processor consists of "blocks", where each block contain a stack of Interaction Networks (each one known as a "message passing layer") with identical structure but different weights. The node and edge functions are 3 layer MLPs activated with shifted softplus (\(ssp(x)=\ln (0.5e^x + 0.5)\) , [1]}), followed by layer normalisation [2]}. Residual connections on both the nodes and edges are added to each message passing layer. We recurrently apply each block, a process we we call "block iteration". We calculate the number of message passing layers by multiplying the number of blocks by the block size, for example 10 applications of a block of size 10 corresponds to 100 (\(10\times 10\) ) message passing layers.

[2]

https://openalex.org/W3037932933

1b9f8ffb-4dee-4e63-9146-adcf9f672a3d

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[1]

https://openalex.org/W2854445389

08de4475-f3eb-4937-85e2-c73fd1fc082c

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[2]

https://openalex.org/W2890208753

c8989c27-f64c-4371-908f-7b9a8d54bb3b

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[4]

https://openalex.org/W2963430173

a5482cb5-c76d-471b-ab37-9619845f13f3

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[5]

https://openalex.org/W2900152462

94d7b642-d49d-423a-b3b6-76349f867e75

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[6]

https://openalex.org/W2786019934

45119972-e403-4b67-a703-dab9a4614348

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[7]

https://openalex.org/W2889347284

89255aa9-01c1-4d1e-8fe6-71752bae0391

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[9]

https://openalex.org/W3025660841

4c794f34-046f-4a06-a4e4-21bd5fd8c69e

The ability to adapt flexibly and efficiently to novel environments is one of the most distinctive and compelling features of the human mind. It has been suggested that humans do so by learning internal models which not only contain abstract representations of the world, but also encode generalisable, structural relationships within the environment [1]}. This latter aspect, it is conjectured, is what allows humans to adapt efficiently and selectively. Recent efforts have been made to mimic this kind of representation in machine learning. World models [2]} aim to capture the dynamics of an environment by distilling past experience into a parametric predictive model. Advances in latent variable models have enabled the learning of world models in a compact latent space [2]}, [4]}, [5]}, [6]}, [7]} from high-dimensional observations such as images. Whilst these models have enabled agents to act in complex environments via planning [5]}, [9]} or learning parametric policies [10]}, [2]}, structurally adapting to changes in the environment remains a significant challenge. The consequence of this limitation is particularly pronounced when deploying learning agents to environments, where distribution shifts occur. As such, we argue that it is beneficial to build structural world models that afford modular and efficient adaptation, and that causal modeling offers a tantalising prospect to discover such structure from observations.

[10]

https://openalex.org/W2995298643

51fde515-9424-45c5-9124-91d45fd4d3a1

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[1]

https://openalex.org/W3135588948

283f40a6-fcb1-423c-bae1-4d045cd5a4a0

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[3]

https://openalex.org/W2914607694

01556cf1-3fc5-4feb-9c40-2f5f0894f1b3

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[4]

https://openalex.org/W2144020560

27adb62f-9203-4ec5-a581-fca74fb2e168

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[5]

https://openalex.org/W1905064697

3304177f-2e17-4cdd-b12b-9dc87b8ac588

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[6]

https://openalex.org/W879220392

911c1c88-36d0-4c96-afae-1a3d5402b709

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[7]

https://openalex.org/W1524326598

3688fb92-0d60-4ac6-a71b-8376d066b415

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[8]

https://openalex.org/W3133236490

05023560-4e8c-4204-b99c-9fc499d03b57

Causality plays a central role in understanding distribution changes, which can be modelled as causal interventions [1]}. The Sparse Mechanism Shift hypothesis [1]}, [3]} (SMS) states that naturally occurring shifts in the data distribution can be attributed to sparse and local changes in the causal generative process. This implies that many causal mechanisms remain invariant across domains [4]}, [5]}, [6]}. In this light, learning a causal model of the environment enables agents to reason about distribution shifts and to exploit the invariance of learnt causal mechanisms across different environments. Hence, we posit that world models with a causal structure can facilitate modular transfer of knowledge. To date, however, methods for causal discovery [7]}, [8]}, [9]}, [10]}, [11]} require access to abstract causal variables to learn causal models from data. These are not typically available in the context of world model learning, where we wish to operate directly on high-dimensional observations.

[9]

https://openalex.org/W2801890059