15954145
Exhaustive and Efficient Constraint Propagation: A Graph-Based Learning Approach and Its Applications
This paper presents a novel pairwise constraint propagation approach by decomposing the challenging constraint propagation problem into a set of independent semi-supervised classification subproblems which can be solved in quadratic time using label propagation based on $$k$$-nearest neighbor graphs. Considering that this time cost is proportional to the number of all possible pairwise constraints, our approach actually provides an efficient solution for exhaustively propagating pairwise constraints throughout the entire dataset. The resulting exhaustive set of propagated pairwise constraints are further used to adjust the similarity matrix for constrained spectral clustering. Other than the traditional constraint propagation on single-source data, our approach is also extended to more challenging constraint propagation on multi-source data where each pairwise constraint is defined over a pair of data points from different sources. This multi-source constraint propagation has an important application to cross-modal multimedia retrieval. Extensive results have shown the superior performance of our approach.
8968350
From Instance-level Constraints to Space-Level Constraints: Making the Most of Prior Knowledge in Data Clustering
We present an improved method for clustering in the presence of very limited supervisory information, given as pairwise instance constraints. By allowing instance-level constraints to have space-level inductive implications, we are able to successfully incorporate constraints for a wide range of data set types. Our method greatly improves on the previously studied constrained k-means algorithm, generally requiring less than half as many constraints to achieve a given accuracy on a range of real-world data, while also being more robust when over-constrained. We additionally discuss an active learning algorithm which increases the value of constraints even further.
25991764
Semi-supervised graph clustering: a kernel approach
Semi-supervised clustering algorithms aim to improve clustering results using limited supervision. The supervision is generally given as pairwise constraints; such constraints are natural for graphs, yet most semi-supervised clustering algorithms are designed for data represented as vectors. In this paper, we unify vector-based and graph-based approaches. We show that a recently-proposed objective function for semi-supervised clustering based on Hidden Markov Random Fields, with squared Euclidean distance and a certain class of constraint penalty functions, can be expressed as a special case of the weighted kernel k-means objective. A recent theoretical connection between kernel k-means and several graph clustering objectives enables us to perform semi-supervised clustering of data given either as vectors or as a graph. For vector data, the kernel approach also enables us to find clusters with non-linear boundaries in the input data space. Furthermore, we show that recent work on spectral learning (Kamvar et al., 2003) may be viewed as a special case of our formulation. We empirically show that our algorithm is able to outperform current state-of-the-art semi-supervised algorithms on both vector-based and graph-based data sets.
2643381
Distance Metric Learning with Application to Clustering with Side-Information
Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they consider "similar." For instance, we may ask them to provide examples. In this paper, we present an algorithm that, given examples of similar (and, if desired, dissimilar) pairs of points in ℝn, learns a distance metric over ℝn that respects these relationships. Our method is based on posing metric learning as a convex optimization problem, which allows us to give efficient, local-optima-free algorithms. We also demonstrate empirically that the learned metrics can be used to significantly improve clustering performance.
9096902
Semi-supervised sparse metric learning using alternating linearization optimization
In plenty of scenarios, data can be represented as vectors and then mathematically abstracted as points in a Euclidean space. Because a great number of machine learning and data mining applications need proximity measures over data, a simple and universal distance metric is desirable, and metric learning methods have been explored to produce sensible distance measures consistent with data relationship. However, most existing methods suffer from limited labeled data and expensive training. In this paper, we address these two issues through employing abundant unlabeled data and pursuing sparsity of metrics, resulting in a novel metric learning approach called semi-supervised sparse metric learning. Two important contributions of our approach are: 1) it propagates scarce prior affinities between data to the global scope and incorporates the full affinities into the metric learning; and 2) it uses an efficient alternating linearization method to directly optimize the sparse metric. Compared with conventional methods, ours can effectively take advantage of semi-supervision and automatically discover the sparse metric structure underlying input data patterns. We demonstrate the efficacy of the proposed approach with extensive experiments carried out on six datasets, obtaining clear performance gains over the state-of-the-arts.
18764978
On Spectral Clustering: Analysis and an algorithm
Despite many empirical successes of spectral clustering methods— algorithms that cluster points using eigenvectors of matrices derived from the data—there are several unresolved issues. First. there are a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well. We also show surprisingly good experimental results on a number of challenging clustering problems.
15482324
Star Shape Prior for Graph-Cut Image Segmentation
In recent years, segmentation with graph cuts is increasingly used for a variety of applications, such as photo/video editing, medical image processing, etc. One of the most common applications of graph cut segmentation is extracting an object of interest from its background. If there is any knowledge about the object shape (i.e. a shape prior), incorporating this knowledge helps to achieve a more robust segmentation. In this paper, we show how to implement a star shape prior into graph cut segmentation. This is a generic shape prior, i.e. it is not specific to any particular object, but rather applies to a wide class of objects, in particular to convex objects. Our major assumption is that the center of the star shape is known, for example, it can be provided by the user. The star shape prior has an additional important benefit - it allows an inclusion of a term in the objective function which encourages a longer object boundary. This helps to alleviate the bias of a graph cut towards shorter segmentation boundaries. In fact, we show that in many cases, with this new term we can achieve an accurate object segmentation with only a single pixel, the center of the object, provided by the user, which is rarely possible with standard graph cut interactive segmentation.
242900
Constrained spectral clustering through affinity propagation
Pairwise constraints specify whether or not two samples should be in one cluster. Although it has been successful to incorporate them into traditional clustering methods, such as K-means, little progress has been made in combining them with spectral clustering. The major challenge in designing an effective constrained spectral clustering is a sensible combination of the scarce pairwise constraints with the original affinity matrix. We propose to combine the two sources of affinity by propagating the pairwise constraints information over the original affinity matrix. Our method has a Gaussian process interpretation and results in a closed-form expression for the new affinity matrix. Experiments show it outperforms state-of-the-art constrained clustering methods in getting good clusterings with fewer constraints, and yields good image segmentation with user-specified pairwise constraints.
5756169
Pairwise constraint propagation by semidefinite programming for semi-supervised classification
We consider the general problem of learning from both pairwise constraints and unlabeled data. The pairwise constraints specify whether two objects belong to the same class or not, known as the must-link constraints and the cannot-link constraints. We propose to learn a mapping that is smooth over the data graph and maps the data onto a unit hypersphere, where two must-link objects are mapped to the same point while two cannot-link objects are mapped to be orthogonal. We show that such a mapping can be achieved by formulating a semidefinite programming problem, which is convex and can be solved globally. Our approach can effectively propagate pairwise constraints to the whole data set. It can be directly applied to multi-class classification and can handle data labels, pairwise constraints, or a mixture of them in a unified framework. Promising experimental results are presented for classification tasks on a variety of synthetic and real data sets.
14179884
Segmentation given partial grouping constraints
We consider data clustering problems where partial grouping is known a priori. We formulate such biased grouping problems as a constrained optimization problem, where structural properties of the data define the goodness of a grouping and partial grouping cues define the feasibility of a grouping. We enforce grouping smoothness and fairness on labeled data points so that sparse partial grouping information can be effectively propagated to the unlabeled data. Considering the normalized cuts criterion in particular, our formulation leads to a constrained eigenvalue problem. By generalizing the Rayleigh-Ritz theorem to projected matrices, we find the global optimum in the relaxed continuous domain by eigendecomposition, from which a near-global optimum to the discrete labeling problem can be obtained effectively. We apply our method to real image segmentation problems, where partial grouping priors can often be derived based on a crude spatial attentional map that binds places with common salient features or focuses on expected object locations. We demonstrate not only that it is possible to integrate both image structures and priors in a single grouping process, but also that objects can be segregated from the background without specific object knowledge.
206747107
Constrained Spectral Clustering via Exhaustive and Efficient Constraint Propagation
This paper presents an exhaustive and efficient constraint propagation approach to exploiting pairwise constraints for spectral clustering. Since traditional label propagation techniques cannot be readily generalized to propagate pairwise constraints, we tackle the constraint propagation problem inversely by decomposing it to a set of independent label propagation subproblems which are further solved in quadratic time using semi-supervised learning based on k-nearest neighbors graphs. Since this time complexity is proportional to the number of all possible pairwise constraints, our approach gives a computationally efficient solution for exhaustively propagating pairwise constraint throughout the entire dataset. The resulting exhaustive set of propagated pairwise constraints are then used to adjust the weight (or similarity) matrix for spectral clustering. It is worth noting that this paper first clearly shows how pairwise constraints are propagated independently and then accumulated into a conciliatory closed-form solution. Experimental results on real-life datasets demonstrate that our approach to constrained spectral clustering outperforms the state-of-the-art techniques.
508435
Learning with Local and Global Consistency
We consider the general problem of learning from labeled and unlabeled data, which is often called semi-supervised learning or transductive inference. A principled approach to semi-supervised learning is to design a classifying function which is sufficiently smooth with respect to the intrinsic structure collectively revealed by known labeled and unlabeled points. We present a simple algorithm to obtain such a smooth solution. Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data.
12957010
Solution of the matrix equation AX + XB = C [F4]
and Canada) or $18.00 (elsewhere). If the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S. only). All orders are to be prepaid with checks payable to ACM Algorithms. The algorithm is re corded as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape. We will supply the algorithm at a density of 800 B.P.I. if requested. The cards for the algorithm are sequenced starting at 10 and incremented by 10. The sequence number is right justified in colums 80. Although we will make every attempt to insure that the algorithm conforms to the description printed here, we cannot guarantee it, nor can we guarantee that the algorithm is correct.-L.D.F. Descdption The following programs are a collection of Fortran IV sub-routines to solve the matrix equation AX-.}-XB = C (1) where A, B, and C are real matrices of dimensions m X m, n X n, and m X n, respectively. Additional subroutines permit the efficient solution of the equation ArX + xa = C, (2) where C is symmetric. Equation (1) has applications to the direct solution of discrete Poisson equations [2]. It is well known that (1) has a unique solution if and only if the One proof of the result amounts to constructing the solution from complete systems of eigenvalues and eigenvectors of A and B, when they exist. This technique has been proposed as a computational method (e.g. see [1 ]); however, it is unstable when the eigensystem is ill conditioned. The method proposed here is based on the Schur reduction to triangular form by orthogonal similarity transformations. Equation (1) is solved as follows. The matrix A is reduced to lower real Schur form A' by an orthogonal similarity transformation U; that is A is reduced to the real, block lower triangular form.
118884461
Lyapunov Matrix Equation in System Stability and Control
Part 1 Introduction: stability of linear systems variance of linear stochastic systems quadratic performance measure book organization. Part 2 Continuous algebraic Lyapunov equation: explicit solutions solution sounds numerical solutions. Part 3 Discrete algebraic Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions. Part 4 Differential and difference Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions singularly perturbed and weakly coupled systems coupled differential equations. Part 5 Algebraic Lyapunov equation with small parameters: singularly perturbed continuous Lyapunov equation weakly coupled continuous Lyapunov equation singularly perturbed discrete systems recursive methods for weakly coupled discrete systems. Part 6 Robustness and sensitivity of the Lyapunov equation: stability robustness sensitivity of algebraic Lyapunov equation. Part 7 Iterative methods and parallel algorithms: Smith's algorithm ADI iterative method SOR iterative method parallel algorithms parallel algorithms for coupled Lyapunov equations. Part 8 Lyapunov iterations: Kleinman algorithm for Riccati equation Lyapunov iterations for jump linear systems Lyapunov iterations for Nash differential games Lyapunov iterations for output feedback control. Part 9 Concluding remarks: Sylvester equations related topics applications. Appendix: matrix inequalities.
5882785
Design of Multimodal Dissimilarity Spaces for Retrieval of Video Documents
The paper proposes a novel representation space for multimodal information, enabling fast and efficient retrieval of video data. We suggest describing the documents not directly by selected multimodal features (audio, visual, or text) but rather by considering cross-document similarities relative to their multimodal characteristics. This idea leads us to propose a particular form of dissimilarity space that is adapted to the asymmetric classification problem and, in turn, to the query-by-example and relevance feedback paradigm, widely used in information retrieval. Based on the proposed dissimilarity space, we then define various strategies to fuse modalities through a kernel-based learning approach. The problem of automatic kernel setting to adapt the learning process to the queries is also discussed. The properties of our strategies are studied and validated on artificial data. In a second phase, a large annotated video corpus (i.e., TRECVID '05) indexed by visual, audio, and text features is considered to evaluate the overall performance of the dissimilarity space and fusion strategies. The obtained results confirm the validity of the proposed approach for the representation and retrieval of multimodal information in a real-time framework.
162180
Multimodal Video Indexing: A Review of the State-of-the-art
Efficient and effective handling of video documents depends on the availability of indexes. Manual indexing is unfeasible for large video collections. In this paper we survey several methods aiming at automating this time and resource consuming process. Good reviews on single modality based video indexing have appeared in literature. Effective indexing, however, requires a multimodal approach in which either the most appropriate modality is selected or the different modalities are used in collaborative fashion. Therefore, instead of separately treating the different information sources involved, and their specific algorithms, we focus on the similarities and differences between the modalities. To that end we put forward a unifying and multimodal framework, which views a video document from the perspective of its author. This framework forms the guiding principle for identifying index types, for which automatic methods are found in literature. It furthermore forms the basis for categorizing these different methods.
14674271
Clustering with Soft and Group Constraints
Several clustering algorithms equipped with pairwise hard constraints between data points are known to improve the accuracy of clustering solutions. We develop a new clustering algorithm that extends mixture clustering in the presence of (i) soft constraints, and (ii) group-level constraints. Soft constraints can reflect the uncertainty associated with a priori knowledge about pairs of points that should or should not belong to the same cluster, while group-level constraints can capture larger building blocks of the target partition when afforded by the side information. Assuming that the data points are generated by a mixture of Gaussians, we derive the EM algorithm to estimate the parameters of different clusters. Empirical study demonstrates that the use of soft constraints results in superior data partitions normally unattainable without constraints. Further, the solutions are more robust when the hard constraints may be incorrect.
1052837
Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions
An approach to semi-supervised learning is proposed that is based on a Gaussian random field model. Labeled and unlabeled data are represented as vertices in a weighted graph, with edge weights encoding the similarity between instances. The learning problem is then formulated in terms of a Gaussian random field on this graph, where the mean of the field is characterized in terms of harmonic functions, and is efficiently obtained using matrix methods or belief propagation. The resulting learning algorithms have intimate connections with random walks, electric networks, and spectral graph theory. We discuss methods to incorporate class priors and the predictions of classifiers obtained by supervised learning. We also propose a method of parameter learning by entropy minimization, and show the algorithm's ability to perform feature selection. Promising experimental results are presented for synthetic data, digit classification, and text classification tasks.
207155154
A probabilistic framework for semi-supervised clustering
Unsupervised clustering can be significantly improved using supervision in the form of pairwise constraints, i.e., pairs of instances labeled as belonging to same or different clusters. In recent years, a number of algorithms have been proposed for enhancing clustering quality by employing such supervision. Such methods use the constraints to either modify the objective function, or to learn the distance measure. We propose a probabilistic model for semi-supervised clustering based on Hidden Markov Random Fields (HMRFs) that provides a principled framework for incorporating supervision into prototype-based clustering. The model generalizes a previous approach that combines constraints and Euclidean distance learning, and allows the use of a broad range of clustering distortion measures, including Bregman divergences (e.g., Euclidean distance and I-divergence) and directional similarity measures (e.g., cosine similarity). We present an algorithm that performs partitional semi-supervised clustering of data by minimizing an objective function derived from the posterior energy of the HMRF model. Experimental results on several text data sets demonstrate the advantages of the proposed framework.
3028284
Automatic Linguistic Indexing of Pictures by a Statistical Modeling Approach
Automatic linguistic indexing of pictures is an important but highly challenging problem for researchers in computer vision and content-based image retrieval. In this paper, we introduce a statistical modeling approach to this problem. Categorized images are used to train a dictionary of hundreds of statistical models each representing a concept. Images of any given concept are regarded as instances of a stochastic process that characterizes the concept. To measure the extent of association between an image and the textual description of a concept, the likelihood of the occurrence of the image based on the characterizing stochastic process is computed. A high likelihood indicates a strong association. In our experimental implementation, we focus on a particular group of stochastic processes, that is, the two-dimensional multiresolution hidden Markov models (2D MHMMs). We implemented and tested our ALIP (Automatic Linguistic Indexing of Pictures) system on a photographic image database of 600 different concepts, each with about 40 training images. The system is evaluated quantitatively using more than 4,600 images outside the training database and compared with a random annotation scheme. Experiments have demonstrated the good accuracy of the system and its high potential in linguistic indexing of photographic images.
37042540
Contextual Kernel and Spectral Methods for Learning the Semantics of Images
This paper presents contextual kernel and spectral methods for learning the semantics of images that allow us to automatically annotate an image with keywords. First, to exploit the context of visual words within images for automatic image annotation, we define a novel spatial string kernel to quantify the similarity between images. Specifically, we represent each image as a 2-D sequence of visual words and measure the similarity between two 2-D sequences using the shared occurrences of s -length 1-D subsequences by decomposing each 2-D sequence into two orthogonal 1-D sequences. Based on our proposed spatial string kernel, we further formulate automatic image annotation as a contextual keyword propagation problem, which can be solved very efficiently by linear programming. Unlike the traditional relevance models that treat each keyword independently, the proposed contextual kernel method for keyword propagation takes into account the semantic context of annotation keywords and propagates multiple keywords simultaneously. Significantly, this type of semantic context can also be incorporated into spectral embedding for refining the annotations of images predicted by keyword propagation. Experiments on three standard image datasets demonstrate that our contextual kernel and spectral methods can achieve significantly better results than the state of the art.
122166830
Relations Between Two Sets of Variates
Concepts of correlation and regression may be applied not only to ordinary one-dimensional variates but also to variates of two or more dimensions. Marksmen side by side firing simultaneous shots at targets, so that the deviations are in part due to independent individual errors and in part to common causes such as wind, provide a familiar introduction to the theory of correlation; but only the correlation of the horizontal components is ordinarily discussed, whereas the complex consisting of horizontal and vertical deviations may be even more interesting. The wind at two places may be compared, using both components of the velocity in each place. A fluctuating vector is thus matched at each moment with another fluctuating vector. The study of individual differences in mental and physical traits calls for a detailed study of the relations between sets of correlated variates. For example the scores on a number of mental tests may be compared with physical measurements on the same persons. The questions then arise of determining the number and nature of the independent relations of mind and body shown by these data to exist, and of extracting from the multiplicity of correlations in the system suitable characterizations of these independent relations. As another example, the inheritance of intelligence in rats might be studied by applying not one but s different mental tests to N mothers and to a daughter of each
189915041
Comparing partitions
The problem of comparing two different partitions of a finite set of objects reappears continually in the clustering literature. We begin by reviewing a well-known measure of partition correspondence often attributed to Rand (1971), discuss the issue of correcting this index for chance, and note that a recent normalization strategy developed by Morey and Agresti (1984) and adopted by others (e.g., Miligan and Cooper 1985) is based on an incorrect assumption. Then, the general problem of comparing partitions is approached indirectly by assessing the congruence of two proximity matrices using a simple cross-product measure. They are generated from corresponding partitions using various scoring rules. Special cases derivable include traditionally familiar statistics and/or ones tailored to weight certain object pairs differentially. Finally, we propose a measure based on the comparison of object triples having the advantage of a probabilistic interpretation in addition to being corrected for chance (i.e., assuming a constant value under a reasonable null hypothesis) and bounded between ±1.
7962793
From Comparing Clusterings to Combining Clusterings
This paper presents a fast simulated annealing framework for combining multiple clusterings (i.e. clustering ensemble) based on some measures of agreement between partitions, which are originally used to compare two clusterings (the obtained clustering vs. a ground truth clustering) for the evaluation of a clustering algorithm. Though we can follow a greedy strategy to optimize these measures as objective functions of clustering ensemble, some local optima may be obtained and simultaneously the computational cost is too large. To avoid the local optima, we then consider a simulated annealing optimization scheme that operates through single label changes. Moreover, for these measures between partitions based on the relationship (joined or separated) of pairs of objects such as Rand index, we can update them incrementally for each label change, which makes sure the simulated annealing optimization scheme is computationally feasible. The simulation and real-life experiments then demonstrate that the proposed framework can achieve superior results.
11664336
Modeling the Shape of the Scene: A Holistic Representation of the Spatial Envelope
In this paper, we propose a computational model of the recognition of real world scenes that bypasses the segmentation and the processing of individual objects or regions. The procedure is based on a very low dimensional representation of the scene, that we term the Spatial Envelope. We propose a set of perceptual dimensions (naturalness, openness, roughness, expansion, ruggedness) that represent the dominant spatial structure of a scene. Then, we show that these dimensions may be reliably estimated using spectral and coarsely localized information. The model generates a multidimensional space in which scenes sharing membership in semantic categories (e.g., streets, highways, coasts) are projected closed together. The performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category.
1260607
Scene Classification Via pLSA
Given a set of images of scenes containing multiple object categories (e.g. grass, roads, buildings) our objective is to discover these objects in each image in an unsupervised manner, and to use this object distribution to perform scene classification. We achieve this discovery using probabilistic Latent Semantic Analysis (pLSA), a generative model from the statistical text literature, here applied to a bag of visual words representation for each image. The scene classification on the object distribution is carried out by a k-nearest neighbour classifier. 
 
We investigate the classification performance under changes in the visual vocabulary and number of latent topics learnt, and develop a novel vocabulary using colour SIFT descriptors. Classification performance is compared to the supervised approaches of Vogel & Schiele [19] and Oliva & Torralba [11], and the semi-supervised approach of Fei Fei & Perona [3] using their own datasets and testing protocols. In all cases the combination of (unsupervised) pLSA followed by (supervised) nearest neighbour classification achieves superior results. We show applications of this method to image retrieval with relevance feedback and to scene classification in videos.
17301592
Bridging the Gap: Query by Semantic Example
A combination of query-by-visual-example (QBVE) and semantic retrieval (SR), denoted as query-by-semantic-example (QBSE), is proposed. Images are labeled with respect to a vocabulary of visual concepts, as is usual in SR. Each image is then represented by a vector, referred to as a semantic multinomial, of posterior concept probabilities. Retrieval is based on the query-by-example paradigm: the user provides a query image, for which 1) a semantic multinomial is computed and 2) matched to those in the database. QBSE is shown to have two main properties of interest, one mostly practical and the other philosophical. From a practical standpoint, because it inherits the generalization ability of SR inside the space of known visual concepts (referred to as the semantic space) but performs much better outside of it, QBSE produces retrieval systems that are more accurate than what was previously possible. Philosophically, because it allows a direct comparison of visual and semantic representations under a common query paradigm, QBSE enables the design of experiments that explicitly test the value of semantic representations for image retrieval. An implementation of QBSE under the minimum probability of error (MPE) retrieval framework, previously applied with success to both QBVE and SR, is proposed, and used to demonstrate the two properties. In particular, an extensive objective comparison of QBSE with QBVE is presented, showing that the former significantly outperforms the latter both inside and outside the semantic space. By carefully controlling the structure of the semantic space, it is also shown that this improvement can only be attributed to the semantic nature of the representation on which QBSE is based.