Increasing subsequences, matrix loci, and Viennot shadows
- URL: http://arxiv.org/abs/2306.08718v2
- Date: Tue, 29 Oct 2024 01:21:04 GMT
- Title: Increasing subsequences, matrix loci, and Viennot shadows
- Authors: Brendon Rhoades,
- Abstract summary: We show that the quotient $mathbbF[mathbfx_n times n]/I_n$ admits a standard monomial basis.
We also calculate the structure of $mathbbF[mathbfx_n times n]/I_n$ as a graded $mathfrakS_n times mathfrakS_n$-module.
- Score: 0.0
- License:
- Abstract: Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring in these variables over a field $\mathbb{F}$. We study the ideal $I_n \subseteq \mathbb{F}[\mathbf{x}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot's shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ is the generating function of permutations in $\mathfrak{S}_n$ by the length of their longest increasing subsequence. Along the way, we describe a `shadow junta' basis of the vector space of $k$-local permutation statistics. We also calculate the structure of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ as a graded $\mathfrak{S}_n \times \mathfrak{S}_n$-module.
Related papers
- The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$.
As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
arXiv Detail & Related papers (2024-10-26T06:21:42Z) - A class of ternary codes with few weights [0.0]
In this paper, we investigate a ternary code $mathcalC$ of length $n$, defined by $mathcalC$ := (textTr) := (textTr(dx), dots, dots, d_n$.
Using recent results on explicit evaluations of exponential sums, we determine the Weil bound, and techniques, we show that the dual code of $mathcalC$ is optimal with respect to the Hamming bound.
arXiv Detail & Related papers (2024-10-05T16:15:50Z) - In-depth Analysis of Low-rank Matrix Factorisation in a Federated Setting [21.002519159190538]
We analyze a distributed algorithm to compute a low-rank matrix factorization on $N$ clients.
We obtain a global $mathbfV$ in $mathbbRd times r$ common to all clients and a local $mathbfUi$ in $mathbbRn_itimes r$.
arXiv Detail & Related papers (2024-09-13T12:28:42Z) - Efficient Continual Finite-Sum Minimization [52.5238287567572]
We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization.
Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires.
We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
arXiv Detail & Related papers (2024-06-07T08:26:31Z) - Quantum charges of harmonic oscillators [55.2480439325792]
We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$.
We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
arXiv Detail & Related papers (2024-04-02T09:16:18Z) - Provably learning a multi-head attention layer [55.2904547651831]
Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models.
In this work, we initiate the study of provably learning a multi-head attention layer from random examples.
We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
arXiv Detail & Related papers (2024-02-06T15:39:09Z) - A Fast Optimization View: Reformulating Single Layer Attention in LLM
Based on Tensor and SVM Trick, and Solving It in Matrix Multiplication Time [7.613259578185218]
We focus on giving a provable guarantee for the one-layer attention network objective function $L(X,Y).
In a multi-layer LLM network, the matrix $B in mathbbRn times d2$ can be viewed as the output of a layer.
We provide an iterative algorithm to train loss function $L(X,Y)$ up $epsilon$ that runs in $widetildeO( (cal T_mathrmmat(n,d) + d
arXiv Detail & Related papers (2023-09-14T04:23:40Z) - Learning a Single Neuron with Adversarial Label Noise via Gradient
Descent [50.659479930171585]
We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations.
The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
arXiv Detail & Related papers (2022-06-17T17:55:43Z) - Spectral properties of sample covariance matrices arising from random
matrices with independent non identically distributed columns [50.053491972003656]
It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$.
Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
arXiv Detail & Related papers (2021-09-06T14:21:43Z) - Bulk-boundary asymptotic equivalence of two strict deformation
quantizations [0.0]
The existence of a strict deformation quantization of $X_k=S(M_k(mathbbC))$ has been proven by both authors and K. Landsman citeLMV.
A similar result is known for the symplectic manifold $S2subsetmathbbR3$.
arXiv Detail & Related papers (2020-05-09T12:03:18Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.