While considerable theoretical progress has been devoted to the study of metric and similarity learning, the generalization mystery is still missing. In this paper, we study the generalization performance of metric and similarity learning by leveraging the specific structure of the true metric (the target function). Specifically, by deriving the explicit form of the true metric for metric and similarity learning with the hinge loss, we construct a structured deep ReLU neural network as an approximation of the true metric, whose approximation ability relies on the network complexity. Here, the network complexity corresponds to the depth, the number of nonzero weights and the computation units of the network. Consider the hypothesis space which consists of the structured deep ReLU networks, we develop the excess generalization error bounds for a metric and similarity learning problem by estimating the approximation error and the estimation error carefully. An optimal excess risk rate is derived by choosing the proper capacity of the constructed hypothesis space. To the best of our knowledge, this is the first-ever-known generalization analysis providing the excess generalization error for metric and similarity learning. In addition, we investigate the properties of the true metric of metric and similarity learning with general losses.

Explanations in Computer Vision are often desired, but most Deep Neural Networks can only provide saliency maps with questionable faithfulness. Self-Explaining Neural Networks (SENN) extract interpretable concepts with fidelity, diversity, and grounding to combine them linearly for decision-making. While they can explain what was recognized, initial realizations lack accuracy and general applicability. We propose the Quantized-Self-Explaining Neural Network "Q-SENN". Q-SENN satisfies or exceeds the desiderata of SENN while being applicable to more complex datasets and maintaining most or all of the accuracy of an uninterpretable baseline model, outperforming previous work in all considered metrics. Q-SENN describes the relationship between every class and feature as either positive, negative or neutral instead of an arbitrary number of possible relations, enforcing more binary humanfriendly features. Since every class is assigned just 5 interpretable features on average, Q-SENN shows convincing local and global interpretability. Additionally, we propose a feature alignment method, capable of aligning learned features with human language-based concepts without additional supervision. Thus, what is learned can be more easily verbalized.

Convex clustering is a modern method with both hierarchical and $k$-means clustering characteristics. Although convex clustering can capture complex clustering structures hidden in data, the existing convex clustering algorithms are not scalable to large data sets with sample sizes greater than several thousands. Moreover, it is known that convex clustering sometimes fails to produce a complete hierarchical clustering structure. This issue arises if clusters split up or the minimum number of possible clusters is larger than the desired number of clusters. In this paper, we propose convex clustering through majorization-minimization (CCMM) -- an iterative algorithm that uses cluster fusions and a highly efficient updating scheme derived using diagonal majorization. Additionally, we explore different strategies to ensure that the hierarchical clustering structure terminates in a single cluster. With a current desktop computer, CCMM efficiently solves convex clustering problems featuring over one million objects in seven-dimensional space, achieving a solution time of 51 seconds on average.

This article measures how sparsity can make neural networks more robust to membership inference attacks. The obtained empirical results show that sparsity improves the privacy of the network, while preserving comparable performances on the task at hand. This empirical study completes and extends existing literature.

In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms.

The unstructured sparsity after pruning poses a challenge to the efficient implementation of deep learning models in existing regular architectures like systolic arrays. On the other hand, coarse-grained structured pruning is suitable for implementation in regular architectures but tends to have higher accuracy loss than unstructured pruning when the pruned models are of the same size. In this work, we propose a model compression method based on a novel weight permutation scheme to fully exploit the fine-grained weight sparsity in the hardware design. Through permutation, the optimal arrangement of the weight matrix is obtained, and the sparse weight matrix is further compressed to a small and dense format to make full use of the hardware resources. Two pruning granularities are explored. In addition to the unstructured weight pruning, we also propose a more fine-grained subword-level pruning to further improve the compression performance. Compared to the state-of-the-art works, the matrix compression rate is significantly improved from 5.88x to 14.13x. As a result, the throughput and energy efficiency are improved by 2.75 and 1.86 times, respectively.

A BSTRACT In probabilistic classification, a discriminative model based on Gaussian mixture exhibits flexible fitting capability. Nevertheless, it is difficult to determine the number of components. We propose a sparse classifier based on a discriminative Gaussian mixture model (GMM), which is named sparse discriminative Gaussian mixture (SDGM). In the SDGM, a GMM-based discriminative model is trained by sparse Bayesian learning. This learning algorithm improves the generalization capability by obtaining a sparse solution and automatically determines the number of components by removing redundant components. The SDGM can be embedded into neural networks (NNs) such as convolutional NNs and can be trained in an end-to-end manner. Experimental results indicated that the proposed method prevented overfitting by obtaining sparsity. Furthermore, we demonstrated that the proposed method outperformed a fully connected layer with the softmax function in certain cases when it was used as the last layer of a deep NN. 1 I NTRODUCTION In supervised classification, probabilistic classification is an approach that assigns a class label c to an input sample x by estimating the posterior probability P (c x).

Low bit-width integer weights and activations are very important for efficient inference, especially with respect to lower power consumption. We propose Monte Carlo methods to quantize the weights and activations of pre-trained neural networks without any re-training. By performing importance sampling we obtain quantized low bit-width integer values from full-precision weights and activations. The precision, sparsity, and complexity are easily configurable by the amount of sampling performed. Our approach, called Monte Carlo Quantization (MCQ), is linear in both time and space, with the resulting quantized, sparse networks showing minimal accuracy loss when compared to the original full-precision networks. Our method either outperforms or achieves competitive results on multiple benchmarks compared to previous quantization methods that do require additional training.

We study the necessary and sufficient complexity of ReLU neural networks-in terms of depth and number of weights-required for approximating classifier functions in an $L^2$-sense. As a model, we consider the set $\mathcal{E}^\beta (\mathbb{R}^d)$ of possibly discontinuous piecewise $C^\beta$ functions $f : [-1/2, 1/2]^d \to \mathbb{R}$, where the different 'smooth regions' of $f$ are separated by $C^\beta$ hypersurfaces. For given dimension $d \geq 2$, regularity $\beta > 0$, and accuracy $\varepsilon > 0$, we construct ReLU neural networks that approximate functions from $\mathcal{E}^\beta(\mathbb{R}^d)$ up to an $L^2$ error of $\varepsilon$. The constructed networks have a fixed number of layers, depending only on $d$ and $\beta$ and they have $O(\varepsilon^{-2(d-1)/\beta})$ many nonzero weights, which we prove to be optimal. In addition to the optimality in terms of the number of weights, we show that in order to achieve this optimal approximation rate, one needs ReLU networks of a certain minimal depth. Precisely, for piecewise $C^\beta(\mathbb{R}^d)$ functions, this minimal depth is given-up to a multiplicative constant-by $\beta/d$. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions. Finally, we analyze approximation in high-dimensional spaces where the function $f$ to be approximated can be factorized into a smooth dimension reducing feature map $\tau$ and classifier function $g$-defined on a low-dimensional feature space-as $f = g \circ \tau$. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.

Encoder-decoder GANs architectures (e.g., BiGAN and ALI) seek to add an inference mechanism to the GANs setup, consisting of a small encoder deep net that maps data-points to their succinct encodings. The intuition is that being forced to train an encoder alongside the usual generator forces the system to learn meaningful mappings from the code to the data-point and vice-versa, which should improve the learning of the target distribution and ameliorate mode-collapse. It should also yield meaningful codes that are useful as features for downstream tasks. The current paper shows rigorously that even on real-life distributions of images, the encode-decoder GAN training objectives (a) cannot prevent mode collapse; i.e. the objective can be near-optimal even when the generated distribution has low and finite support (b) cannot prevent learning meaningless codes for data -- essentially white noise. Thus if encoder-decoder GANs do indeed work then it must be due to reasons as yet not understood, since the training objective can be low even for meaningless solutions.