Related papers: Minimal Inversions in Integer Matrices of Fixed RS…
The task of predicting missing entries of a matrix, from a subset of known entries, is known as \textit{matrix completion}. In today's data-driven world, data completion is essential whether it is the main goal or a pre-processing step.…
In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be…
We continue work begun in \cite{ptab} which introduced \emph{perforated tableaux} as a combinatorial model for crystals of type $A_{n-1}$, emphasizing connections to the classical Robinson-Schensted-Knuth (RSK) correspondence and Lusztig…
Shuffled linear regression (SLR) seeks to estimate latent features through a linear transformation, complicated by unknown permutations in the measurement dimensions. This problem extends traditional least-squares (LS) and Least Absolute…
We present a simple formula to update the pseudoinverse of a full-rank rectangular matrix that undergoes a low-rank modification, and demonstrate its utility for solving least squares problems. The resulting algorithm can be dramatically…
We give an affine analogue of the Robison-Schensted-Knuth (RSK) correspondence, which generalizes the affine Robinson-Schensted correspondence by Chmutov-Pylyavskyy-Yudovina. The affine RSK map sends a generalized affine permutation of…
Short-depth algorithms are crucial for reducing computational error on near-term quantum computers, for which decoherence and gate infidelity remain important issues. Here we present a machine-learning approach for discovering such…
If $\alpha \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\alpha$ is $\Phi(\alpha) = \{(i, j) \, | \, 1 \leq i < j \leq n, \alpha(i) > \alpha(j)\}$. We describe all $r$-tuples $\alpha_1, \alpha_2, \ldots, \alpha_r…
Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence…
It is well known that the minimum $\ell_2$-norm solution of the convex LASSO model, say $\mathbf{x}_{\star}$, is a continuous piecewise linear function of the regularization parameter $\lambda$, and its signed sparsity pattern is constant…
We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It…
We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\lambda/\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape…
We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants.…
We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform…
We develop and analyze a broad family of stochastic/randomized algorithms for inverting a matrix. We also develop specialized variants maintaining symmetry or positive definiteness of the iterates. All methods in the family converge…
A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a…
Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if…
Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple…
An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed…
A Robinson similarity matrix is a symmetric matrix where the entry values on all rows and columns increase toward the diagonal. Decompose the Robinson matrix into the sum of k {0, 1}-matrices, then these k {0, 1}-matrices are the adjacency…