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Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Although prohibitively large, such matrices are often approximately low rank. We propose an algorithm that exploits this structure to obtain a low rank decomposition of any matrix A as A≈LR, where L and R are the low rank factors. The total number of elements in L and R can be significantly less than that in A. Furthermore, the entries of L and R are quantized to low precision formats −− compressing A by giving us a low rank and low precision factorization. Our algorithm first computes an approximate basis of the range space of A by randomly sketching its columns, followed by a quantization of the vectors constituting this basis. It then computes approximate projections of the columns of A onto this quantized basis. We derive upper bounds on the approximation error of our algorithm, and analyze the impact of target rank and quantization bit-budget. The tradeoff between compression ratio and approximation accuracy allows for flexibility in choosing these parameters based on specific application requirements. We empirically demonstrate the efficacy of our algorithm in image compression, nearest neighbor classification of image and text embeddings, and compressing the layers of LlaMa-7b. Our results illustrate that we can achieve compression ratios as aggressive as one bit per matrix coordinate, all while surpassing or maintaining the performance of traditional compression techniques. 

We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the nonconvex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provides a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks. 

Three features are crucial for sequential forecasting and generation models: tractability, expressiveness, and theoretical backing. While neural autoregressive models are relatively tractable and offer powerful predictive and generative capabilities, they often have complex optimization landscapes, and their theoretical properties are not well understood. To address these issues, we present convex formulations of autoregressive models with one hidden layer. Specifically, we prove an exact equivalence between these models and constrained, regularized logistic regression by using semi-infinite duality to embed the data matrix onto a higher dimensional space and introducing inequality constraints. To make this formulation tractable, we approximate the constraints using a hinge loss or drop them altogether. Furthermore, we demonstrate faster training and competitive performance of these implementations compared to their neural network counterparts on a variety of data sets. Consequently, we introduce techniques to derive tractable, expressive, and theoretically-interpretable models that are nearly equivalent to neural autoregressive models. 

Fisher’s Linear Discriminant Analysis (FLDA) is a statistical analysis method that linearly embeds data points to a lower dimensional space to maximize a discrimination criterion such that the variance between classes is maximized while the variance within classes is minimized. We introduce a natural extension of FLDA that employs neural networks, called Neural Fisher Discriminant Analysis (NFDA). This method finds the optimal two-layer neural network that embeds data points to optimize the same discrimination criterion. We use tools from convex optimization to transform the optimal neural network embedding problem into a convex problem. The resulting problem is easy to interpret and solve to global optimality. We evaluate the method’s performance on synthetic and real datasets. 

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although post-training quantization is well-studied, training optimal quantized NNs involves combinatorial non-convex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in two-layer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to global optimality provably in polynomial time in all relevant parameters via tight semidefinite relaxations. We present numerical examples to illustrate the effectiveness of our method. 

Chronic stress has been associated with a variety of pathophysiological risks including developing mental illness. Conversely, appropriate stress management, can be used to foster mental wellness proactively. Yet, there is no existing method that accurately and objectively monitors stress. With recent advances in electronic-skin (e-skin) and wearable technologies, it is possible to design devices that continuously measure physiological parameters linked to chronic stress and other mental health and wellness conditions. However, the design approach should be different from conventional wearables due to considerations like signal-to-noise ratio and the risk of stigmatization. Here, we present a multi-part study that combines user-centered design with engineering-centered data collection to inform future design efforts. To assess human factors, we conducted an n=24 participant design probe study that examined perceptions of an e-skin for mental health and wellness as well as preferred wear locations. We complement this with an n=10 and n=16 participant data collection study to measure physiological signals at several potential wear locations. By balancing human factors and biosignals, we conclude that the upper arm and forearm are optimal wear locations.