Graph Spectral Regularization For Neural Network Interpretability

Deep neural networks can learn meaningful representations of data . However , these representations are hard to interpret . For example , visualizing a latent layer is generally only possible for at most three dimensions . Neural networks are able to learn and benefit from much higher dimensional representations but these are not visually interpretable because nodes have arbitrary ordering within a layer . Here , we utilize the ability of the human observer to identify patterns in structured representations to visualize higher dimensions . To do so , we propose a class of regularizations we call Graph Spectral Regularizations that impose graphstructure on latent layers . This is achieved by treating activations as signals on a predefined graph and constraining those activations using graph filters , such as low pass and wavelet-like filters . This framework allows for any kind of graph as well as filter to achieve a wide range of structured regularizations depending on the inference needs of the data . First , we show a synthetic example that the graph-structured layer can reveal topological features of the data . Next , we show that a smoothing regularization can impose semantically consistent ordering of nodes when applied to capsule nets . Further , we show that the graph-structured layer , using wavelet-like spatially localized filters , can form localized receptive fields for improved image and biomedical data interpretation . In other words , the mapping between latent layer , neurons and the output space becomes clear due to the localization of the activations . Finally , we show that when structured as a grid , the representations create coherent images that allow for image-processing techniques such as convolutions . 1 INTRODUCTION . Neural networks have revolutionized many areas of machine learning including image and natural language processing . However , one of the major challenges for neural networks is that they are still black boxes to the user . It is not quite clear how network internals map from inputs to outputs or how to interpret the features learned by the nodes . This is mainly because the features are not constrained to have specific structure or characteristics . Existing regularizations constrain the learned code to have certain properties . However , they are not designed to specifically aid in interpretation of the latent encoding . For example , L1 regularization induces sparsity in the activations but does not impose specific structure between dimensions . Here , we introduce a new class of regularizations called Graph Spectral Regularizations that result in activations that are filtered on a predefined graph . We define specific members of this class for applications . First , we introduce a ( graph ) Laplacian smoothing regularization which enforces smoothly varying activations while at the same time reconstructing the data . This regularization is useful for learning features with specific topologies . For instance , we show that on a clusterstructured topology where features correspond to hierarchical cluster structure in the data it reflects the abstract grouping of features . We also show it is useful for inducing feature consistency between nodes of capsule networks ( Sabour et al. , 2017 ) . The graph regularization semantically aligns the features such that they appear in the same order in each capsule . When trained on MNIST digits , we find that each of our 10 capsules consisting of 16 nodes encodes the same transformation ( rotation , scale , skew , etc ) of a particular digit in the same node . While the Laplacian smoothing regularizations is useful in the context where the features of the data have a recognizable topology , often we don ’ t know the explicit structure of the data . Instead , we would like to extract the topology of the data itself . Thus , we design a filter that encourages the graph structure layer to learn data-shape features . We achieve this by using a spatially localized , Gaussian filter to localize the activations for any particular data point . We ensure that only one of a dictionary of localized filters is chosen as the activation via a spectral bottleneck layer preceding the graph-structured layer . We show that spatially-localized filter regularizations are useful for detecting circular and linear topologies of data that are not immediately reflected by the observed features . We also explore a biological system – a single-cell protein expression dataset depicting T cell development in the thymus – that has continuous progression structure . The graph structured layer ( with a ring graph ) reveals the data to have a Y-shaped topology reflecting the bifurcation into CD4 ( regulatory ) and CD8 ( cytotoxic ) T cells , confirming known T cell biology . Finally , we show that the graph-structured layer , when imposing a 2D grid , creates a “ pseudo ” image that can be analyzed by convolution layers . We show that such re-encoded images of MNIST digits have localized receptive fields that can be used for classification and visual interpretability . Interestingly , we find that the convolution obviates the need for a spectral bottleneck as the convolution and max pooling themselves may provide that function . Our contributions are as follows : • A framework for imposing graph structure on latent layers using graph spectral regularizations . • A Laplacian graph smoothing regularization and its applications in learning feature smooth- ness and consistency . • Spatially localized graph regularizations using a spectral bottleneck based on a dictionary of Gaussian Kernels and its application in recognizing data topology . • Applications of graph spectral regularizations , natural and biological datasets to demon- strate feature interpretability and data topology . The rest of this paper is organized as follows . We first define graph structured layers and two techniques utilizing this layer in Section 2 . Then we present experiments demonstrating improved interpretability in Section 3 . Finally , we wrap up with conclusions in Section 4 . 2 GRAPH-STRUCTURED LAYER . Consider a given layer in an artificial neural network , and let V = { ν1 , . . . , νn } be the neurons in this layer . These neurons can essentially be regarded as functions of inputs to the neural network , and map each input to an n dimensional vector ν ( x ) = [ ν1 ( x ) , . . . , νn ( x ) ] T ∈ Rn . Typically , no particular structure is being directly imposed on the range of ν beyond general notions of bounded Lp norm ( e.g. , Euclidean L2 norm or L1 norm as a proxy for sparsity ) . This is clear with fully connected layers , but even with convolutional ones that introduce some relation between neurons within each layer , there is still no clear topological structure for the representations obtained by the entire layer as a whole . Indeed , convolutional layers typically learn multiple channels ( or filters ) from input signals , and while neurons within each channel can be organized in spatiotemporal coordinates , there are no imposed relations between the different channels . Since many applications of neural networks essentially focus on supervised learning tasks ( whether predictive or generative ) , where hidden layer neurons are only used as intermediate computational units , their unstructured nature is not considered an important issue . However , when using neural networks for unsupervised and exploratory tasks , in which the hidden layers are treated as latent data representations , their lack of structure makes the interpretation of the resulting representations challenging if not impossible . To address this challenge , autoencoders and similar unsupervised deep models typically restrict their bottleneck layer to have two or three neurons , thus mapping data points to R2 or R3 , where the entire data can be visualized as a 2D or 3D cloud of points . While this approach is useful for getting a general understanding of structures ( e.g. , clustering or trends ) in the data , it significantly limits the amount of information that can be captured by such latent representations . Furthermore , it underutilizes the ability of a human observer to identify rich patterns in 2D displays . Indeed , human perception in natural settings is not tuned to observe particle clouds , but rather to recognize shapes , identify textures , and assess relative sizes of objects . Such elements are often used by visualization techniques in various fields , such as TreeMap ( Shneiderman , 1992 ) , audio spectrograms ( Flanagan , 1972 ) , and even classic box plots and histograms ( Tukey , 1977 ) . Here , we propose a new approach for producing human interpretable patterns in the latent representation obtained by hidden layers of neural networks . To this end , we impose a graph topology on the neurons in V , which enables us to control spectral properties ( e.g. , regularity and locality ) of latent representations obtained by them to produce recognizable patterns . Formally , we consider the neurons V as vertices of a graph with weighted edges between them defined by an n× n adjacency matrixW . Then , for any given input x to the network , we consider the activations [ ν1 ( x ) , . . . , νn ( x ) ] as a signal ν over this neuron graph . We propose here three ways to utilize this graph signal structure of a neural network layer . First , by considering graph signal processing notions , we can define spectral regularizations that control the regularity or smoothness of learned signals , as explained in Section 2.1 . Second , by utilizing sparse graph wavelet dictionaries , we can define a new type of bottleneck that forces autoencoder middle layers to encode inputs by sparse set of localized dictionary atoms ( e.g. , as done in compressive sensing and dictionary learning ) , as explained in Section 2.2 . Finally , by utilizing simple graph structures , such as a ring graph or a 2D lattice we can force downstream layers to be convolutional layers , regardless of whether the input of the network was structured or not . Section 3 demonstrates the utility of each of these design choices for several data exploration applications , in both supervised and unsupervised settings . 2.1 SPECTRAL GRAPH REGULARIZATION . The graph structure defined by the weighted adjacency matrix W naturally provides a notion of locality , based on local graph neighborhoods , or geosesic distances on the graph . However , recent works on graph signal processing ( e.g. , Shuman et al. , 2013 , and references therein ) also explored spectral notions provided by graph structures , which extend and generalize traditional signal processing and harmonic analysis notions . These notions are based on the definition of a graph Laplacian as L = D −W , D = diag ( deg ( ν1 ) , . . . , deg ( νn ) ) , deg ( νj ) = ‖W ( j , · ) ‖1 , j = 1 , . . . , n , which provides ( either directly or via proper normalization ) a discrete version of well-studied manifold Laplace operators from geometric harmonic analysis ( e.g. , Belkin & Niyogi , 2002 ; Coifman & Lafon , 2006 ) . Then , the eigenvectors ψ0 , . . . , ψn−1 of L ( indexed by convention in ascending order of the corresponding eigenvalues ) are considered graph Fourier harmonics . These can be shown to converge to discrete Fourier harmonics when considering a ring graph , in which case their associated eigenvalues , 0 = λ0 ≤ · · · ≤ λn−1 , correspond to squared frequencies . This understanding gives rise to the graph Fourier transform defined as a linear transformation characterized by the matrix Ψ whose columns are the graph harmonics { ψj } n−1j=0 . When applied to a neuron-activation signal ν in our setting we get its graph Fourier coefficients as ν̂ = ΨTν , where ν̂ is associated with the graph ( squared ) frequency λj . Similarly , the inverse Fourier transform is also defined via Ψ , as it can be verified that Ψν̂ = ΨΨTν = ν , since the graph Laplacian ( for all graphs considered in this work ) yields a full orthonormal set of eigenvectors , and thus Ψ is an orthogonal matrix . The graph Fourier transform allows us to consider the activations ν ( x ) for a given input x of the network in two domains . First , in their original form , these activations form a signal over the neuron graph , which can essentially be considered as a function of individual neurons , with ν ( x ) [ i ] = νi ( x ) , i = 1 , . . . , n , which we refer to as the neuron-domain representation . Alternatively , we can consider this signal in the spectral domain , via its Fourier coefficients , as a function ν̂ ( x ) [ j ] , j = 0 , . . . , n−1 , of graph-harmonic indices . This allows us to pose a new set of regularizations that are defined in the spectral domain , rather than in the neuron domain , in order to directly enforce spectral properties of the neuron activation signal . We note that one of the most popular regularization traditionally used in deep learning is the L2 regularization , which essentially adds the squared Euclidean norm of the activations in the neuron domain ( i.e. , ‖ν ( x ) ‖22 ) as another term in the main optimization target for gradient descent . Such regularization encourages the activations to all have equivalently small values , essentially providing a global notion of smoothness that is then balanced with other loss terms but typically provides stability to the optimization process ( Goodfellow et al. , 2016 ) . Since it can be verified that the graph Fourier transform is energy preserving ( i.e. , due to the orthonormality of graph harmonics ) , this regularization can equivalently be considered in the spectral domain by using ‖ν̂ ( x ) ‖22 instead of ‖ν ( x ) ‖22 . However , unlike the neuron domain , in the spectral domain we can associate with each element in ν̂ ( x ) a frequencial interpretation given by { λj } nj=0 , and therefore we can also generalize the spectral L2 regularization to be weighted by functions of these squared-frequencies . Namely , given weights µj = µ ( λj ) , j = 0 , . . . , n − 1 , for some function µ , we define the spectral L2 ( µ ) regularization as adding the term ‖ν̂ ( x ) ‖2L2 ( µ ) = n−1∑ j=0 µj ( ν̂ ( x ) [ j ] ) 2 = [ ν̂ ( x ) ] T µ ν̂ ( x ) , where µ = diag ( µ0 , . . . , µn−1 ) , to the optimization loss of the neural network . Notice that such weighting can not be directly defined in the neuron domain , as individual neurons are not associated with any a priori interpretation . Instead , to apply the spectral regularization in the neuron domain , we consider the matrix form on the RHS , and by combining it together with the definition of the graph Fourier transform we can write the L2 ( µ ) spectral regularization term in the neuron domain as the quadratic form [ ν ( x ) ] TF ( µ ) ν ( x ) , where F ( µ ) = ΨµΨT is a matrix that is independent of the network input x , and can thus be directly computed in advance from the neuron graph structure , and the predetermined spectral weights in µ . Finally , while in this work we focus on the L2 ( µ ) form of spectral regularizations , in general other weighted norms ( e.g. , L1 ( µ ) or more generally Lp ( µ ) , p ≥ 1 ) can also be used to define spectral regularizaions based on based on the harmonic structure induced by the neuron graph . Laplacian smoothing regularization : we now focus on given class of L2 ( µ ) spectral regularizations with nonnegative weights , i.e. , µ ≥ 0 . In such cases , the chosen weights determine which harmonic bands to penalize , and by how much , in the resulting regularization . Therefore , these spectral regularization enable to guide the latent representation provided by the graph-structured layer towards , or away from , certain harmonic patterns . In particular , this enables us to encourage smooth latent representations ( over the neuron graph sense ) by using weights that penalize high frequency . A natural choice for such weights is to simply set them by the identity µ ( λj ) = λj , which results in F ( µ ) = L. We refer to this regularization as Laplacian smoothing , as the resulting quadratic loss term for it is [ ν ( x ) ] TLν ( x ) , and demonstrate its utility for producing interpretable latent representations in hidden layers of neural networks in Section 3 .

Summary: Authors present a novel regularizer to impose graph structure upon hidden layers of a neural Network. The intuition is that Neural Networks has typically  symmetric computation among different channels in one layer. Due to the lack of order, visually inspecting the hidden representation is not feasible. By adding edges one can impose a structure upon nodes in one layer and add for example a Laplacian regularizer rather than simple L2 norm regularizer to force the activations to follow the imposed structure.