Almost-Orthogonality in Lp Spaces: A Case Study with Grok

• URL: http://arxiv.org/abs/2605.05192v1
• Date: Wed, 06 May 2026 17:54:51 GMT
• Title: Almost-Orthogonality in Lp Spaces: A Case Study with Grok
• Abstract summary: Carbery proposed the following sharpened form of triangle inequality for many functions.<n>Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.
• Score: 9.384449875311857
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p}, \] where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound \[ \Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p}, \] where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

Related papers

• Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere [0.0]
  We prove an isospectral version of Lieb-Solovej inequality.<n>Spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_$.
  arXiv  Detail & Related papers  (2026-08-12T16:46:26Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• 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)
• Tight Lower Bounds under Asymmetric High-Order Hölder Smoothness and Uniform Convexity [6.309677398331863]
  We establish worst-case oracle complexities of $Omegaleft( left( fracHsigmaright)frac23(p+nu)$ in terms of the optimality gap.<n>Our results match the corresponding upper bounds in this general setting.
  arXiv  Detail & Related papers  (2024-09-16T23:17:33Z)
• 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 Algorithms and Lower Bounds for Finite-Sum Optimization [22.076317220348145]
  We give a quantum algorithm with complexity $tildeObig(n+sqrtd+sqrtell/mubig)$, improving the classical tight bound $tildeThetabig(n+sqrtnell/mubig)$. We also prove a quantum lower bound $tildeOmega(n+n3/4(ell/mu)1/4)$ when $d$ is large enough.
  arXiv  Detail & Related papers  (2024-06-05T07:13:52Z)
• Dimension Independent Disentanglers from Unentanglement and Applications [55.86191108738564]
  We construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show that to capture NEXP, it suffices to have unentangled proofs of the form $| psi rangle = sqrta | sqrt1-a | psi_+ rangle where $| psi_+ rangle has non-negative amplitudes.
  arXiv  Detail & Related papers  (2024-02-23T12:22:03Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• On the continuous Zauner conjecture [0.0]
  In this paper we prove that for any $t in [-frac1d2-1, frac1d+1] setminus0$ the equality $textebr(Phi_t)=d2$ is equivalent to the existence of a pair of informationally complete unit norm tight frames.
  arXiv  Detail & Related papers  (2021-12-11T00:14:35Z)
• Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes [12.640283469603357]
  We study the computational complexity of computing the normalizing constant. We show that $sum_Sdet(bf A_S,S)p$ exactly for every (fixed) positive even integer $p$ is UP-hard and Mod$_3$P-hard.
  arXiv  Detail & Related papers  (2021-11-28T14:08:25Z)
• On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
  We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
  arXiv  Detail & Related papers  (2020-02-09T03:39:46Z)
• Some convergent results for Backtracking Gradient Descent method on Banach spaces [0.0]
  bf Theorem. Let $X$ be a Banach space and $f:Xrightarrow mathbbR$ be a $C2$ function. Let $mathcalC$ be the set of critical points of $f$.
  arXiv  Detail & Related papers  (2020-01-16T12:49:42Z)

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.