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This paper proposes an optimization-based method to learn the singular value decomposition (SVD) of a compact operator with ordered singular functions. The proposed objective function is based on Schmidt’s low-rank approximation theorem (1907) that characterizes a truncated SVD as a solution minimizing the mean squared error, accompanied with a technique called nesting to learn the ordered structure. When the optimization space is parameterized by neural networks, we refer to the proposed method as NeuralSVD. The implementation does not require sophisticated optimization tricks unlike existing approaches. 

We study the problem of extracting biometric informa- tion of individuals by looking at shadows of objects cast on diffuse surfaces. We show that the biometric information leakage from shadows can be sufficient for reliable identity inference under representative scenarios via a maximum like- lihood analysis. We then develop a learning-based method that demonstrates this phenomenon in real settings, exploit- ing the subtle cues in the shadows that are the source of the leakage without requiring any labeled real data. In par- ticular, our approach relies on building synthetic scenes composed of 3D face models obtained from a single photo- graph of each identity. We transfer what we learn from the synthetic data to the real data using domain adaptation in a completely unsupervised way. Our model is able to general- ize well to the real domain and is robust to several variations in the scenes. We report high classification accuracies in an identity classification task that takes place in a scene with unknown geometry and occluding objects. 

Generalization error bounds are essential to understanding machine learning algorithms. This paper presents novel expected generalization error upper bounds based on the average joint distribution between the output hypothesis and each input training sample. Multiple generalization error upper bounds based on different information measures are provided, including Wasserstein distance, total variation distance, KL divergence, and Jensen-Shannon divergence. Due to the convexity of the information measures, the proposed bounds in terms of Wasserstein distance and total variation distance are shown to be tighter than their counterparts based on individual samples in the literature. An example is provided to demonstrate the tightness of the proposed generalization error bounds. 

With the unprecedented performance achieved by deep learning, it is commonly believed that deep neural networks (DNNs) attempt to extract informative features for learning tasks. To formalize this intuition, we apply the local information geometric analysis and establish an information-theoretic framework for feature selection, which demonstrates the information-theoretic optimality of DNN features. Moreover, we conduct a quantitative analysis to characterize the impact of network structure on the feature extraction process of DNNs. Our investigation naturally leads to a performance metric for evaluating the effectiveness of extracted features, called the H-score, which illustrates the connection between the practical training process of DNNs and the information-theoretic framework. Finally, we validate our theoretical results by experimental designs on synthesized data and the ImageNet dataset. 

We generalize the information bottleneck (IB) and privacy funnel (PF) problems by introducing the notion of a sensitive attribute, which arises in a growing number of applications. In this generalization, we seek to construct representations of observations that are maximally (or minimally) informative about a target variable, while also satisfying constraints with respect to a variable corresponding to the sensitive attribute. In the Gaussian and discrete settings, we show that by suitably approximating the Kullback-Liebler (KL) divergence defining traditional Shannon mutual information, the generalized IB and PF problems can be formulated as semi-definite programs (SDPs), and thus efficiently solved, which is important in applications of high-dimensional inference. We validate our algorithms on synthetic data and demonstrate their use in imposing fairness in machine learning on real data as an illustrative application.