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tokens: These levels of abstraction are learned in an unsupervised (i.e., autonomous) fashion from a large catalog of input spectra. Once these features are learned from the data, a second, supervised learning stage is used to assign the learned features to their correct labels (e.g., H 2 O, CH 4 , etc.). Neural networks are now commonly used in complex classification tasks such as image recognition (e.g., Krizhevsky et al. 2012 ; Liu et al. 2014 ; Wang et al. 2016 ; Shen et al. 2015 ), speech and music recognition (e.g., Hung et al. 2005 ; Jaitly Hinton 2011 ; Zhang Wu 2013 ; Pradeep Kumaraswamy 2014 ), biology (e.g., Head-Gordon Stillinger 1993 ; Plebe 2007 ; Wu McLarty 2012 ; Spencer et al. 2015 ), and are finding increased use in the classification of galaxies and cosmology (e.g., Collister Lahav 2004 ; Agarwal et al. 2012 , 2014 ; Reis et al. 2012 ; Karpenka et al. 2013 ; Dieleman et al. 2015 ; du Buisson et al. 2015 ; Ellison et al. 2015 ; Huertas-Company et al. 2015 ). Whereas an in-depth derivation of DBNs is beyond the scope of this paper, we will briefly outline its underlying architecture and implementation. We refer the interested reader to Bengio ( 2009 ), Hinton ( 2012 ), and Fischer Igel ( 2014 ) for detailed derivations. 2.1. Restricted Boltzmann Machines Figure 1 shows a schematic of the the deep-belief network. The multi-layer DBN can be constructed from several Restricted Boltzmann Machines (Freund Haussler 1992 ; Bishop 2006 ; Le Roux Bengio 2008 ; Bengio 2009 , 2012 ; Lee et al. 2011a ; Hinton 2012 ; Montavon et al. 2012 ; Fischer Igel 2014 ) with the addition of a logistic regression layer at the top of the network. The RBM is a two-layer neural network able to learn the underlying probability distribution over its set of input values. It represents a particular kind of Markov Random Field (Davison 2008 ) consisting of one layer of binary or Gaussian stochastic visible units (the input data) and one layer of binary stochastic hidden units. In RBMs, all of the hidden units are connected to all of the visible units but have no intra-layer dependence. Hence, all of the hidden units given the visible units are statistically independent and we can write the probability of all visible units given all hidden units and vice versa as the product of the individual probabilities:where and are the column vectors of the visible and hidden units, respectively, and i and j are their corresponding indices. We now want to find a configuration of the hidden layers, , that allows us to reconstruct the input, , with minimal error. Since and are factorial, we can write the activation functions of the individual visible and hidden binary units aswhere ς is the sigmoid functionAssuming that both the visible and hidden units are binary, RBMs assign an energy term for each configuration of and :where and are the bias vectors for the visible and hidden units, respectively, and W is a matrix of connection weights between and . The probability over