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Abstract-Deep neural networks have achieved near-human accuracy levels in various types of classification and prediction tasks including images, text, speech, and video data. However, the networks continue to be treated mostly as black-box function approximators, mapping a given input to a classification output. The next step in this human-machine evolutionary processincorporating these networks into mission critical processes such as medical diagnosis, planning and control -requires a level of trust association with the machine output.Typically, statistical metrics are used to quantify the uncertainty of an output. However, the notion of trust also depends on the visibility that a human has into the working of the machine. In other words, the neural network should provide humanunderstandable justifications for its output leading to insights about the inner workings. We call such models as interpretable deep networks.Interpretability is not a monolithic notion. In fact, the subjectivity of an interpretation, due to different levels of human understanding, implies that there must be a multitude of dimensions that together constitute interpretability. In addition, the interpretation itself can be provided either in terms of the lowlevel network parameters, or in terms of input features used by the model. In this paper, we outline some of the dimensions that are useful for model interpretability, and categorize prior work along those dimensions. In the process, we perform a gap analysis of what needs to be done to improve model interpretability.

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The successive subspace learning (SSL) principle was developed and used to design an interpretable learning model, known as the PixelHop method, for image classification in our prior work. Here, we propose an improved PixelHop method and call it Pixel-Hop++. First, to make the PixelHop model size smaller, we decouple a joint spatial-spectral input tensor to multiple spatial tensors (one for each spectral component) under the spatial-spectral separability assumption and perform the Saab transform in a channel-wise manner, called the channel-wise (c/w) Saab transform. Second, by performing this operation from one hop to another successively, we construct a channel-decomposed feature tree whose leaf nodes contain features of one dimension (1D). Third, these 1D features are ranked according to their cross-entropy values, which allows us to select a subset of discriminant features for image classification. In Pixel-Hop++, one can control the learning model size of fine-granularity, offering a flexible tradeoff between the model size and the classification performance. We demonstrate the flexibility of PixelHop++ on MNIST, Fashion MNIST, and CIFAR-10 three datasets.