arXiv subjects

Rafael Chiclana

Publications and source records attributed to Rafael Chiclana.

15 recordsLinked to original sources

arXiv2026

Randomly Permuted Orthogonal Products and Fast Dimension Reduction

We study the effect of random signed permutations on products of orthogonal matrices and their applications to fast dimension reduction. Let $A,B \in \mathbb{R}^{d\times d}$ be orthogonal matrices and let $Σ\in \mathbb{R}^{d\times d}$ be a uniformly random signed permutation matrix. We analyze the random orthogonal matrix \[ U=A ΣB, \] and show that, under mild assumptions on the size of the entries of $A$ and $B$, \[ \max_{i,j=1,\ldots,d} |U_{ij}| =O\left ( \sqrt{\frac{\log d}{d}}\right ) \] with high probability. As an application, we show that ORA, an analogue of the Kac walk in which every update is a $π/4$ rotation, reaches the same maximal entry scale after $O(d\log d)$ updates. This resolves a question of Jain et al. and improves the running time of their construction. We also show that parallel ORA reaches this scale after $O(\log d)$ rounds. We then study the random embedding \[ Φ = \sqrt{\frac{d}{m}}\, P_I U D_{ξ'}, \] where $P_I$ restricts to $m$ coordinates and $ξ'$ is an independent Rademacher vector. We identify two parameters controlling norm preservation and show that, throughout the corresponding admissible range, $Φ$ achieves optimal embedding dimension $m\asymp\varepsilon^{-2}\log(N)$. Finally, we extend the result to structured infinite models, including sparse vectors, low-rank matrices, and finite unions of subspaces.

math.PR↗

arXiv2026

Optimal lower bound for the variance of hitting times for simple random walks on graphs

We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a graph with $n$ vertices, we prove that the variance of the hitting time from a vertex $x$ to a vertex $y$, denoted $τ_y$, is at least of the order $\mathbb{E}_x(τ_y)^2 / \log n$. When the graph is a tree, we show that $n$ can be replaced by the graph's distance between vertices $x$ and $y$.

math.PR↗

arXiv2026

Hereditary Diameter Rigidity in Real $L_1$-Spaces

We prove that hereditary diameter lower bounds are stable under convex combinations in real $L_1$-spaces over arbitrary measure spaces. More precisely, if every weakly open piece of each set has diameter at least $δ>0$, then every finite or countable convex combination has the corresponding hereditary lower bound $δ/4$. In the diameter-two case, the bound is preserved exactly. Consequently, for every topology containing the relative weak topology, the diameter two and strong diameter two properties are equivalent on bounded convex subsets of the unit ball, as are the convex point of continuity property and strong regularity.

math.FA↗

arXiv2026

On Continuous Terminal Embeddings of Sets of Positive Reach

In this paper we prove the existence of Hölder continuous terminal embeddings of any desired $X \subseteq \mathbb{R}^d$ into $\mathbb{R}^{m}$ with $m=\mathcal{O}(\varepsilon^{-2}ω(S_X)^2)$, for arbitrarily small distortion $\varepsilon$, where $ω(S_X)$ denotes the Gaussian width of the unit secants of $X$. More specifically, when $X$ is a finite set we provide terminal embeddings that are locally $\frac{1}{2}$-Hölder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\frac{1}{4}$-Hölder everywhere sufficiently close to $X$ (i.e., within all tubes around $X$ of radius less than $X$'s reach). When $X$ is a compact $d$-dimensional submanifold of $\mathbb{R}^N$, an application of our main results provides terminal embeddings into $\tilde{\mathcal{O}}(d)$-dimensional space that are locally Hölder everywhere sufficiently close to the manifold.

math.OC↗

arXiv2026

The Numerical Index of Two-Dimensional Real $\ell_p$ Spaces

The computation of the numerical index of classical Banach spaces is one of the original problems in the theory. In this paper, we compute the numerical index of two-dimensional real $\ell_p$-spaces for all $p\ge1$. More precisely, we prove that \[ n(\ell_p^2)=v(J), \qquad J= \begin{pmatrix} 0&1 -1&0 \end{pmatrix}, \] confirming the conjectured formula in the two-dimensional real case.

math.FA↗

arXiv2026

Nonconcentration of hitting times for random walks on graphs

We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \[ \operatorname{Var}_x(τ_y)+\mathbb E_xτ_y \ge \frac{(\mathbb E_xτ_y)^2}{1+\log n}, \] with the logarithmic term sharp up to constants. Under a bounded-degree assumption the additive mean term can be removed, giving a variance lower bound depending only on $\mathbb E_xτ_y$ and the graph distance $\dist(x,y)$. We show that this degree assumption is necessary by constructing high-degree graphs with linear mean and bounded variance; the same construction disproves a conjecture of Norris-Peres-Zhai concerning local nonconcentration of hitting times. We also prove a sharper tree estimate, extend the main argument to finite reversible Markov chains, and show that Holroyd's interval conjecture, stated in Norris-Peres-Zhai, fails even for bounded-degree trees.

math.PR↗

arXiv2025

Fast Dimensionality Reduction from $\ell_2$ to $\ell_p$

The Johnson-Lindenstrauss (JL) lemma is a fundamental result in dimensionality reduction, ensuring that any finite set $X \subseteq \mathbb{R}^d$ can be embedded into a lower-dimensional space $\mathbb{R}^k$ while approximately preserving all pairwise Euclidean distances. In recent years, embeddings that preserve Euclidean distances when measured via the $\ell_1$ norm in the target space have received increasing attention due to their relevance in applications such as nearest neighbor search in high dimensions. A recent breakthrough by Dirksen, Mendelson, and Stollenwerk established an optimal $\ell_2 \to \ell_1$ embedding with computational complexity $O(d \log d)$. In this work, we generalize this direction and propose a simple linear embedding from $\ell_2$ to $\ell_p$ for any $p \in [1,2]$ based on a construction of Ailon and Liberty. Our method achieves a reduced runtime of $O(d \log k)$ when $k \leq d^{1/4}$, improving upon prior runtime results when the target dimension is small. Additionally, we show that for \emph{any norm} $\|\cdot\|$ in the target space, any embedding of $(\mathbb{R}^d, \|\cdot\|_2)$ into $(\mathbb{R}^k, \|\cdot\|)$ with distortion $\varepsilon$ generally requires $k = Ω\big(\varepsilon^{-2} \log(\varepsilon^2 n)/\log(1/\varepsilon)\big)$, matching the optimal bound for the $\ell_2$ case up to a logarithmic factor.

math.PR↗

arXiv2025

On Extended Concentration Inequalities for Fast JL Embeddings of Infinite Sets

The Johnson-Lindenstrauss (JL) lemma allows subsets of a high-dimensional space to be embedded into a lower-dimensional space while approximately preserving all pairwise Euclidean distances. This important result has inspired an extensive literature, with a significant portion dedicated to constructing structured random matrices with fast matrix-vector multiplication algorithms that generate such embeddings for finite point sets. In this paper, we briefly consider fast JL embedding matrices for {\it infinite} subsets of $\mathbb{R}^d$. Prior work in this direction such as \cite{oymak2018isometric, mendelson2023column} has focused on constructing fast JL matrices $HD \in \mathbb{R}^{k \times d}$ by multiplying structured matrices with RIP(-like) properties $H \in \mathbb{R}^{k \times d}$ against a random diagonal matrix $D \in \mathbb{R}^{d \times d}$. However, utilizing RIP(-like) matrices $H$ in this fashion necessarily has the unfortunate side effect that the resulting embedding dimension $k$ must depend on the ambient dimension $d$ no matter how simple the infinite set is that one aims to embed. Motivated by this, we explore an alternate strategy for removing this $d$-dependence from $k$ herein: Extending a concentration inequality proven by Ailon and Liberty \cite{Ailon2008fast} in the hope of later utilizing it in a chaining argument to obtain a near-optimal result for infinite sets. %, and $(ii)$ utilizing a simple secondary Gaussian embedding of an initial fast JL embedding of a given infinite set. Though this strategy ultimately fails to provide the near-optimal embedding dimension we seek, along the way we obtain a stronger-than-sub-exponential extension of the concentration inequality in \cite{Ailon2008fast} which may be of independent interest.

cs.DS↗

arXiv2024

On Outer Bi-Lipschitz Extensions of Linear Johnson-Lindenstrauss Embeddings of Subsets of $\mathbb{R}^N$

The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $Φ\in \mathbb{R}^{m \times N}$ with $m=\mathcal{O}(\varepsilon^{-2}\log n)$ such that \[ (1 - \varepsilon) \|x-y\|_2 \leq \|Φx-Φy\|_2 \leq (1+\varepsilon)\| x- y\|_2 \quad \forall\, x, y \in X.\] Herein we consider terminal embedding results which have recently been introduced in the computer science literature as stronger extensions of the Johnson-Lindenstrauss lemma for finite sets. After a short survey of this relatively recent line of work, we extend the theory of terminal embeddings to hold for arbitrary (e.g., infinite) subsets $X \subseteq \mathbb{R}^N$, and then specialize our generalized results to the case where $X$ is a low-dimensional compact submanifold of $\mathbb{R}^N$. In particular, we prove the following generalization of the Johnson-Lindenstrauss lemma: For all $\varepsilon \in (0,1)$ and $X\subseteq\mathbb{R}^N$, there exists a terminal embedding $f: \mathbb{R}^N \longrightarrow \mathbb{R}^{m}$ such that $$(1 - \varepsilon) \| x - y \|_2 \leq \left\| f(x) - f(y) \right\|_2 \leq (1 + \varepsilon) \| x - y \|_2 \quad \forall \, x \in X ~{\rm and}~ \forall \, y \in \mathbb{R}^N.$$ Crucially, we show that the dimension $m$ of the range of $f$ above is optimal up to multiplicative constants, satisfying $m=\mathcal{O}(\varepsilon^{-2} ω^2(S_X))$, where $ω(S_X)$ is the Gaussian width of the set of unit secants of $X$, $S_X=\overline{\{(x-y)/\|x-y\|_2 \colon x \neq y \in X\}}$. Furthermore, our proofs are constructive and yield algorithms for computing a general class of terminal embeddings $f$, an instance of which is demonstrated herein to allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice.

math.MG↗

arXiv2023

A local central limit theorem for random walks on expander graphs

There is a long history of establishing central limit theorems for Markov chains. Quantitative bounds for chains with a spectral gap were proved by Mann and refined later. Recently, rates of convergence for the total variation distance were obtained for random walks on expander graphs, which are often used to generate sequences satisfying desirable pseudorandom properties. We prove a local central limit theorem with an explicit rate of convergence for random walks on expander graphs, and derive an improved bound for the total variation distance.

math.PR↗

arXiv2022

No cutoff in Spherically symmetric trees

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random walks. This answers a question recently proposed by Gantert, Nestoridi, and Schmid. We also show that for lazy simple random walks on finite spherically symmetric trees, hitting times of vertices are (uniformly) non concentrated. Finally, we study the stability of our results under rough isometries.

math.PR↗

arXiv2021

Cantor sets of low density and Lipschitz functions on $C^1$ curves

We characterize the functions $f\colon [0,1] \longrightarrow [0,1]$ for which there exists a measurable set $C\subseteq [0,1]$ of positive measure satisfying $\frac{|C\cap I|}{|I|}<f(|I|)$ for any nontrivial interval $I \subseteq [0,1]$. As an application, we prove that on any $C^1$ curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant.

math.FA↗

arXiv2020

Some stability properties for the Bishop--Phelps--Bollobás property for Lipschitz maps

We study the stability behavior of the Bishop-Phelps-Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very closed to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of $Γ$-flat operators and $ACK$ structure. We get sharper results for the case of Lipschitz compact maps. The behaviour of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above ones about the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.

math.FA↗

arXiv2020

Examples and applications of the density of strongly norm attaining Lipschitz maps

We study the density of the set $\operatorname{SNA}(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which strongly attain their norm (i.e.\ the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that $\operatorname{SNA}(\mathbb T,Y)$ is not dense in ${\mathrm{Lip}}_0(\mathbb T,Y)$ for any Banach space $Y$, where $\mathbb T$ denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e.\ every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold. Next, we construct metric spaces $M$ satisfying that $\operatorname{SNA}(M,Y)$ is dense in ${\mathrm{Lip}}_0(M,Y)$ regardless $Y$ but which contains an isometric copy of $[0,1]$ and so the Lipschitz-free space $\mathcal F(M)$ fails the Radon--Nikodým property, answering in the negative a posed question. Furthermore, an example $M$ can be produced failing all the previously known sufficient conditions to get the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we prove that given a compact metric $M$ which does not contains any isometric copy of $[0,1]$ and a Banach space $Y$, if $\operatorname{SNA}(M,Y)$ is dense, then $\operatorname{SNA}(M,Y)$ actually contains an open dense subset and $B_{\mathcal F(M)}=\overline{\mathrm{co}}(\operatorname{str-exp}(B_{\mathcal F(M)}))$. Further, we show that if $M$ is a boundedly compact metric space for which $\operatorname{SNA}(M,\mathbb R)$ is dense in ${\mathrm{Lip}}_0(M,\mathbb R)$, then the unit ball of the Lipschitz-free space on $M$ is the closed convex hull of its strongly exposed points.

math.FA↗

arXiv2019

The Bishop--Phelps--Bollobás property for Lipschitz maps

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\ the molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the Lip-BPB property for every Banach space $Y$. We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from $(M,\mathbb R)$ to some Banach spaces $Y$ and some results for compact Lipschitz maps are also discussed.

math.FA↗