Related papers: Conditioning of Leverage Scores and Computation by…
Automated short-answer grading (ASAG) remains a challenging task due to the linguistic variability of student responses and the need for nuanced, rubric-aligned partial credit. While Large Language Models (LLMs) offer a promising solution,…
Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic…
The prohibitive sizes of Large Language Models (LLMs) today make it difficult to deploy them on memory-constrained edge devices. This work introduces $\rm CALDERA$ -- a new post-training LLM compression algorithm that harnesses the inherent…
For any matrix A in R^(m x n) of rank \rho, we present a probability distribution over the entries of A (the element-wise leverage scores of equation (2)) that reveals the most influential entries in the matrix. From a theoretical…
For a robust leverage diagnostic in linear regression, Rousseeuw and van Zomeren [1990] proposed using robust distance (Mahalanobis distance computed using robust estimates of location and covariance). However, a design matrix X that…
A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the…
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have…
Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint---known as {\em…
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…
The Kullback-Leibler (KL) divergence is a foundational measure for comparing probability distributions. Yet in multivariate settings, its single value often obscures the underlying reasons for divergence, conflating mismatches in individual…
In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A_1, A_2 and A_3 linearizability…
Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far.…
We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with…
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c…
We characterize the maximal attainable power-size gap in overidentified instrumental variables models with heteroskedastic or autocorrelated (HAC) errors. Using total variation distance and Kraft's theorem, we define the decision theoretic…
Double Machine Learning is often justified by nuisance-rate conditions, yet finite-sample reliability also depends on the conditioning of the orthogonal-score Jacobian. This conditioning is typically assumed rather than tracked. When…
Motivated by the least squares solver Blendenpik, we investigate three strategies for uniform sampling of rows from m x n matrices Q with orthonormal columns. The goal is to determine, with high probability, how many rows are required so…
We analyze numerical properties of Layer-wise relevance propagation (LRP)-type attribution methods by representing them as a product of modified gradient matrices. This representation creates an analogy to matrix multiplications of…
In this note we present a parameterized class of lower triangular matrices. The components of the eigenvectors grow rapidly and will exceed the representational range of any finite number system. The eigenvalues and the eigenvectors are…
There has been significant interest and progress recently in algorithms that solve regression problems involving tall and thin matrices in input sparsity time. These algorithms find shorter equivalent of a n*d matrix where n >> d, which…