Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation

1 INTRODUCTION . Convolutional neural networks ( CNNs ) have achieved great success in various vision tasks Szegedy et al . ( 2016 ) ; Redmon et al . ( 2016 ) ; He et al . ( 2017a ) . The key to such success is the powerful ability of feature representations to input images , where network units1 play a critical role . But impacted by the diverse training deployments and huge hypothesis space , networks even with the same architecture may converge to different minima on a given task . Although units between these networks could present similar function for the same task , yet they may have completely different activation magnitudes . Consequently , this makes it fairly hard to give a general indication of the status for a given network unit with respect to how well features are represented by it from images in the same class . Being rough indicators in practice , magnitude responses of units are usually chosen simply Zhang et al . ( 2018 ) or processed statistically ( such as average mean ) Li et al . ( 2016 ) ; Luo et al . ( 2017 ) based on the idea of matched filtering . However , firstly these indicators are apparently sensitive to rescaling operations in magnitude . If performing a simply rescaling operation to the weights such as the strategy introduced in Neyshabur et al . ( 2015 ) , the results of the network and the function of each unit would all remain unchanged , but these indicators would vary along with the rescaling coefficient . Secondly , as the spatial information in the unit is completely discarded , they could not give discrimination between units with and without random patterns , for example units separately outputted by a well-trained and random-initialized CNN filter . Without a valid indication regarding the mentioned situations , these indicators fail to ensure the universal applicability for units in different network models . In this paper , we attempt to investigate the status of units from a new perspective . Roughly speaking , natural images in the same class have common features , and meanwhile the locations of these features are spatially correlated in global . For effective units , features are picked out and represented by high activation values in the units . And due to the locality nature in feature extraction by convolution , this global spatial pattern between the common features would be preserved synchronously 1Regarding the term unit , a unit is the perceptive node in networks , which generally refers to the activated feature map outputted by a convolutional filter in CNNs . in the counterpart representations in the effective units . In contrast , for ineffective units , being incapability of effectively representing these common features , representations would be in chaos and marks of this pattern is vague . This provides a valid road for performance assessment of individual units , and critically it is rescaling-invariant and universally applicable to any CNN architecture . The investigation of such pattern could naturally lead to topological approaches because knowledge of topological data analysis such as barcodes Ghrist ( 2008 ) provides valuable tools to resolve the intrinsic patterns in raw data . Along this line , firstly we introduce a method for characterizing the spatial pattern of feature representations in units for a single sample by incorporating with the topological tools , and then use information entropy to evaluate the stability of this spatial characterizations for various images sampled from the same class , where we call it feature entropy . In this way , a unit is judged to be effective if its feature entropy is high , otherwise ineffective . In our experiments , we find that feature entropy would gradually decrease as the layer goes deeper and the evolution trends of feature entropy and losses are almost the same during network training . We show that the feature entropy could provide reliable indication of unit status in situations like weight-rescaling and the emergence of random pattern . Finally , we show the value of feature entropy in giving discrimination between networks with different generalization ability by investigating only the training set . 2 RELATED WORKS . One line of research that attracts many researchers is seeking solutions in a way of visualizing what features have learned by the units Zeiler & Fergus ( 2014 ) ; Zhou et al . ( 2014 ) ; Mahendran & Vedaldi ( 2015 ) ; Simonyan et al . ( 2013 ) . Status is generally identified depending on the degree of alignment between the visualized features and the human-visual concepts Bau et al . ( 2017 ) ; Zhou et al . ( 2018a ) ; Bau et al . ( 2020 ) . On the one hand , they meanwhile give excellent visual interpretation of each unit ; on the other hand , it hinders its universal application to arbitrary tasks and models in which units ’ functionalities may be unrecognized to human Wang et al . ( 2020 ) . Another related research trace lies in the field of network pruning , where they concentrate on using simple methods to roughly select less important units within a network . Typical approaches include the L1-Norm of units Luo et al . ( 2017 ) , Average Percentage of Zeros ( APoZ ) in units Hu et al . ( 2016 ) , some sparse-based methods Li et al . ( 2019 ) ; Yoon & Hwang ( 2017 ) , and so on . Despite commonly used in practice , since without a specific processing on units in diverse situations , they are unable to provide a general indication for units in different networks . Besides , Morcos et al . ( 2018 ) introduce the class selectivity from neuroscience to investigate the selectivity over classes for a specific unit , on the basis of calculating the mean units . Alain & Bengio ( 2016 ) propose linear classifier probe , where they report the degree of linear classification of units in intermediate layers could somehow characterize the status of units . Lastly , we would like to discuss some recent works related to topological approaches in deep learning . Naitzat et al . ( 2020 ) demonstrates the superiority of using ReLu activation by studying the changes in Betti numbers of a two-class neural network . Montúfar et al . ( 2020 ) use neural networks to predict the persistent homology features . In Gabrielsson & Carlsson ( 2019 ) , by using barcode , they show the topological structure changes during training which correlates to the generalization of networks . Rieck et al . ( 2018 ) proposes neural persistence , a topological complexity measure of network structure that could give a criterion on early stopping . Guss & Salakhutdinov ( 2018 ) empirically investigates the connection between neural network expressivity and the complexity of dataset in topology . In Hofer et al . ( 2017 ) , topological signatures of data are evaluated and used to improve the classification of shapes . 3 METHOD . In general , input images for a network model are commonly resized to be square for processing . For input image sample I of a given class with size n×n to be represented by a unit U with sizem×m via feature extraction processing f in CNN , we have , f : I → U ( 1 ) For image I , features are specifically arranged , where each feature has an associated spatial location in the image . After perceived by U , features are represented by high activation values at the corresponding locations in the unit . Basically , there are two steps in our assessment of unit performance : firstly , characterizing the spatial pattern hidden in these high activation values in a unit for a single image ; secondly , evaluate the stability of this characterization when giving multiple image samples . 3.1 CHARACTERIZING THE SPATIAL PATTERN OF FEATURE REPRESENTATIONS IN A UNIT . For Ui , j with a grid structure , the location of an element generally refers to its coordinate index ( i , j ) . And intuitively , the spatial pattern hidden in the elements denotes certain regular relationship among their coordinate indices . So , it is natural to model such relationship with graph structure and tackle it with topological tools in the following . Unit and graph We use the edge-weighted graphs Mehmet et al . ( 2019 ) as our basic model and construct the weighted graph G = ( V , E ) from unit Ui , j , where V is the vertex set and E is the edge set . Define the adjacency matrix A of G as follows , A ∈ Rm×m : Ai , j = Ui , j ( 2 ) It should be noted that the individual element of A is the weight of edge in G , which conveys the intensity of corresponding point in the U . A family of undirected graphs G ( v ) with adjacency matrices A ( v ) could be constructed by following the typical implementation of the sublevel set , A ( v ) i , j = 1Ai , j≥a ( v ) ( 3 ) where a ( v ) is the vth value in the descend ordering of elements of A and 1 ( · ) is indicator function . Here , we take the adjustment of A ( v ) = max ( A ( v ) , ( A ( v ) ) T ) to ensure the adjacency matrices A ( v ) of undirected graphs to be symmetric . So G ( v ) = ( V ( v ) , E ( v ) ) is the subgraph of G where V ( v ) = V and E ( v ) ⊂ E only includes the edges whose weights are greater than or equal to a ( v ) . We have the following graph filtration , G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G ( 4 ) To be more specifically , in this sublevel set filtration , it starts with the vertex set , then rank the edge weights from the maximum amax to minimum amin , and let the threshold parameters decrease from amax to amin . At each step , we add the corresponding edges to obtain the threshold subgraph G ( v ) . Fig.2 illustrates the construction of certain subgraph through a toy example . Consider the unit Ui , j . We circle the locations of the top 4 largest elements in Ui , j ( Fig.2A ) . Then the nonzero elements in adjacency matrix A ( 4 ) , { ( 1 , 2 ) , ( 4 , 3 ) , ( 2 , 4 ) , ( 3 , 1 ) } , is located ( Fig.2B ) and corresponding subgraph G ( 4 ) is constructed ( Fig.2C ) . Complex filtration To further reveal the correlation structure in the graphs , they are typically converted into certain kinds of topological objects , where topological invariants are calculated for capturing the high-level abstraction of correlation structure . Here , by following the common method in Horak et al . ( 2009 ) ; Giovanni et al . ( 2013 ) , each graph G ( v ) is converted to simplicial complex ( also called clique complex ) τ ( v ) , as shown in Fig.2D . In this way , we have complex filtration corresponding to graph filtration ( Eq.4 ) . τ ( 1 ) ⊂ τ ( 2 ) ⊂ τ ( 3 ) ⊂ · · · ⊂ τ ( 5 ) This filtration describes the evolution of correlation structure in graph G along with the decreasing of threshold parameter . Fig.3A shows the complex filtration of the previous example ( Fig.2 ) . So far , we have completed the characterization from unit to the topological objects . Other than our strategy , we also discuss other alternative method , which maps the unit to the cubical complex Kaczynski et al . ( 2004 ) . See Appendix for more details . Betti curve and its charaterization Next , kth Betti number Hatcher ( 2002 ) of each element in the complex filtration could be calculated using the typical computational approach of persistent homology Ninna et al . ( 2017 ) . τ ( v ) 7→ β ( τ ( v ) ) ( 6 ) Intuitively , kth Betti number β ( τ ( v ) ) could be regarded as the number of k-dimensional ’ circle ’ s or ’ hole ’ s or some higher order structures in complex τ ( v ) . On the other hand , many meaningful patterns in the unit would lead to the ’ circle ’ s or ’ hole ’ s of complexes in the filtration ( Eq.5 ) , see Fig.2 for illustration . In particular , the number of ’ hole ’ s is typically used as an important quantitative index for featuring such patterns . Hence , the kth Betti numbers β ( τ ( v ) ) , v ∈ { 1 , · · · , n } could be arranged into so called kth Betti curves β ( U , v , k ) for the unit U . Fig.3B shows the 1th Betti curve of filtration in Fig.3A . Once having obtained the Betti curve , one needs to interpret the Betti curve and extract its core characterization . Although there exists many choices of distance between two topological diagrams such as persistence images Adams et al . ( 2017 ) , persistence landscape Bubenik et al . ( 2015 ) and persistence entropyNinna et al . ( 2017 ) , we find that the simple birth time of the Betti curves β ( U , v , k ) is sufficient in this characterization , b ( U , k ) = inf { v|β ( U , v , k ) 6= 0 } ( 7 ) We call b ( U , k ) the birth point . Birth point is the indication of the critical element in complex filtration that begins to carry ” hole ” structure ( Betti number is nonzero ) . It is an important sign that some essential change has occurred in complex filtration , which implies the appearance of regularized spatial pattern of notable components in the unit . Meanwhile , in some cases , no spatial pattern appear in the components in the unit , so β ( U , v , k ) constantly equals to zero , meaning that birth point doesn ’ t exist . In general , this would happen when the unit is unable to give representations for the image , where its values are almost all zeros .

Summary: The paper applies well known methods in topological data analysis for quantitative assessment of neural net unit performance. Specifically, a scheme is proposed wherein spatial patterns of activation in the network units are determined as function of a sequence of activation thresholds, patterns transformed to a graph representations (i.e. successive filtrations) and their corresponding kth betti number, (i.e. number of kth dimensional holes), evolution calculated. In this paper k is set to 1.  For units that the authors term, effective units for a given class and a given input image class population, the analysis is expected to yield to similar birth time of the holes. Thus, a feature entropy is defined on the first birth time in the betti curves computed on a given instance and aggregating the estimated birth times over a population of instances in a given class.  The main results include the illustration of invariance of the calculated feature entropy when one does reweighting / rescaling transformations in the inputs and the illustration of correlation between the choice of hyperparameters, perturbations in data sets, and the feature entropy measure.