# Thomas Fu

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Approximately Optimal Core Shapes for Tensor Decompositions

Mehrdad Ghadiri

Proceedings of the 40th International Conference on Machine Learning (2023), pp. 11237-11254

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This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.
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Sequential Attention for Feature Selection

Taisuke Yasuda

Proceedings of the 11th International Conference on Learning Representations (2023)

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Feature selection is the problem of selecting a subset of features for a machine learning model that maximizes model quality subject to a budget constraint. For neural networks, prior methods, including those based on L1 regularization, attention, and other techniques, typically select the entire feature subset in one evaluation round, ignoring the residual value of features during selection, i.e., the marginal contribution of a feature given that other features have already been selected. We propose a feature selection algorithm called Sequential Attention that achieves state-of-the-art empirical results for neural networks. This algorithm is based on an efficient one-pass implementation of greedy forward selection and uses attention weights at each step as a proxy for feature importance. We give theoretical insights into our algorithm for linear regression by showing that an adaptation to this setting is equivalent to the classical Orthogonal Matching Pursuit (OMP) algorithm, and thus inherits all of its provable guarantees. Our theoretical and empirical analyses offer new explanations towards the effectiveness of attention and its connections to overparameterization, which may be of independent interest.
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Subquadratic Kronecker Regression with Applications to Tensor Decomposition

Mehrdad Ghadiri

Proceedings of the 36th Annual Conference on Neural Information Processing Systems (2022), pp. 28776-28789

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Kronecker regression is a highly-structured least squares problem $\min_{\mathbf{x}} \lVert \mathbf{K}\mathbf{x} - \mathbf{b} \rVert_{2}^2$, where the design matrix $\mathbf{K} = \mathbf{A}^{(1)} \otimes \cdots \otimes \mathbf{A}^{(N)}$ is a Kronecker product of factor matrices. This regression problem arises in each step of the widely-used alternating least squares (ALS) algorithm for computing the Tucker decomposition of a tensor. We present the first \emph{subquadratic-time} algorithm for solving Kronecker regression to a $(1+\varepsilon)$-approximation that avoids the exponential term $O(\varepsilon^{-N})$ in the running time. Our techniques combine leverage score sampling and iterative methods. By extending our approach to block-design matrices where one block is a Kronecker product, we also achieve subquadratic-time algorithms for (1) Kronecker ridge regression and (2) updating the factor matrices of a Tucker decomposition in ALS, which is not a pure Kronecker regression problem, thereby improving the running time of all steps of Tucker ALS. We demonstrate the speed and accuracy of this Kronecker regression algorithm on synthetic data and real-world image tensors.
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Locality-Sensitive Hashing for f-Divergences: Mutual Information Loss and Beyond

Lin Chen

Advances in Neural Information Processing Systems (2019), pp. 10044-10054

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Computing approximate nearest neighbors in high dimensional spaces is a central problem in large-scale data mining with a wide range of applications in machine learning and data science. A popular and effective technique in computing nearest neighbors approximately is the Locality-Sensitive Hashing (LSH) scheme. In this paper, we aim to develop LSH schemes for distance functions that measure the distance between two probability distributions, particularly for f-divergences as well as a generalization to capture mutual information loss.
First, we provide a general framework to design LHS schemes for f-divergence distance functions, and develop LSH schemes for the generalized Jensen-Shannon divergence and triangular discrimination in this framework. We show a two-sided approximation result for approximation of the generalized Jensen-Shannon divergence by the Hellinger distance, which may be of independent interest. Next, we show a general method of reducing the problem of design an LSH scheme for a KreÄ±n kernel (which can be expressed as the difference of two positive definite kernels) to the problem of maximum inner product search. We exemplify this method by applying it to the mutual information loss divergence, due to its several important applications such as model compression.
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Categorical Feature Compression via Submodular Optimization

Lin Chen

International Conference on Machine Learning (2019), pp. 515-523

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In modern machine learning tasks, the presence of categorical features with extremely large vocabularies is a reality. This becomes a bottleneck when using an ML model, which generally grows at least linearly with the vocabulary size, affecting the memory, training and inference costs, as well as overfitting risk. In this work, we seek to compress the vocabulary by maximizing the mutual information between the compressed categorical feature and the target binary labels. We note the relationship of this problem to that of quantization in a discrete memoryless channel, where there exists a quadratic-time algorithm to solve the problem. Unfortunately, such an algorithm does not scale to data sets with massive vocabularies and, in this paper, we develop a distributed quasi-linear O(n log n) algorithm with provable approximation guarantees. We first observe that although entropy is a submodular function, this is not the case for mutual information between a categorical feature and label. To tackle this problem, we define a set function over a different space, which still contains the optimal solution, and prove this function is submodular. We also provide a query oracle to the submodular function that runs in amortized logarithmic time, and is easy to compute in a distributed fashion. Combining these results with a greedy algorithm allows us to achieve a (1-1/e)-approximation in quasi-linear time. Finally, we compare our proposed algorithm to several existing approaches using the large-scale Criteo learning task and demonstrate better performance in retaining mutual information and also verify the learning performance of the compressed vocabulary.
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Greedy Column Subset Selection: New Bounds and Distributed Algorithms

Aditya Bhaskara

Jason Altschuler

ICML (2016) (to appear)

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The problem of matrix column subset selection
has recently attracted a large body of research,
with feature selection serving as one obvious and
important application. Among the techniques
that have been applied to solve this problem, the
greedy algorithm has been shown to be quite
effective in practice. However, our theoretical
guarantees on its performance have not been ex-
plored thoroughly, especially in a distributed set-
ting. In this paper, we study the greedy algorithm
for the column subset selection problem from a
theoretical and empirical perspective and show
its effectiveness in a distributed setting. In par-
ticular, we provide an improved approximation
guarantee for the greedy algorithm, and present
the first distributed implementation of this algo-
rithm with provable approximation factors. We
use the idea of randomized composable core-
sets, developed recently in the context of sub-
modular maximization. Finally, we validate the
effectiveness of this distributed algorithm via an
empirical study.
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