Author

# Francisco Romero Acosta

Bio: Francisco Romero Acosta is an academic researcher. The author has contributed to research in topics: Mathematics & Combinatorics. The author has an hindex of 1, co-authored 6 publications receiving 3 citations.

##### Papers

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12 Aug 2022

TL;DR: In this paper , it was shown that if a compact set E ⊂ R d , d ≥ 3 , has Hausdorﬀ dimension greater than (8 k − 9) (8k − 8) d + k 8 k − 8 , then the set of simplices with vertices in E has nonempty interior.

Abstract: . In this paper we show that if a compact set E ⊂ R d , d ≥ 3 , has Hausdorﬀ dimension greater than (8 k − 9) (8 k − 8) d + k 8 k − 8 , then the set of congruence class of simplices with vertices in E has nonempty interior. By set of congruence class of simplices with vertices in E we mean where 2 ≤ k < d . This result improves our previous work [28] in the sense that we now can obtain a Hausdorﬀ dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of E has nonempty interior when d = 3 as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.

2 citations

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TL;DR: For a compact set E⊂Rd, d ≥ 4, it was shown in this paper that if the Hausdorff dimension of E is larger than 23d+1, then the set of congruence classes of triangles formed by triples of points of E has non-empty interior.

1 citations

20 Dec 2022

TL;DR: The neural manifold hypothesis as mentioned in this paper proposes that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables, thus quantifying their shape within the neural state space.

Abstract: The neural manifold hypothesis postulates that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables. In this work, we combine topological deep generative models and extrinsic Riemannian geometry to introduce a novel approach for studying the structure of neural manifolds. This approach (i) computes an explicit parameterization of the manifolds and (ii) estimates their local extrinsic curvature--hence quantifying their shape within the neural state space. Importantly, we prove that our methodology is invariant with respect to transformations that do not bear meaningful neuroscience information, such as permutation of the order in which neurons are recorded. We show empirically that we correctly estimate the geometry of synthetic manifolds generated from smooth deformations of circles, spheres, and tori, using realistic noise levels. We additionally validate our methodology on simulated and real neural data, and show that we recover geometric structure known to exist in hippocampal place cells. We expect this approach to open new avenues of inquiry into geometric neural correlates of perception and behavior.

06 Oct 2022

TL;DR: In particular, the authors showed that a hypothesis class with finite VC-dimension is PAC-learnable under the assumption that the VC dimension of the hypothesis class is as large as possible.

Abstract: Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.

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TL;DR: In this paper , Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf provide bounds on the minimum number of distinct angles in general position in three dimensions.

Abstract: In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find
the minimum number of distinct distances between pairs of points selected from
any configuration of $n$ points in the plane. The problem has since been
explored along with many variants, including ones that extend it into higher
dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle
problem, which seeks to find point configurations in the plane that minimize
the number of distinct angles. In their recent paper "Distinct Angles in
General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf
use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum
number of distinct angles in the plane in general position, which prohibits
three points on any line or four on any circle.
We consider the question of distinct angles in three dimensions and provide
bounds on the minimum number of distinct angles in general position in this
setting. We focus on pinned variants of the question, and we examine explicit
constructions of point configurations in $\mathbb{R}^3$ which use
self-similarity to minimize the number of distinct angles. Furthermore, we
study a variant of the distinct angles question regarding distinct angle chains
and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$
and $\mathbb{R}^3$.

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03 Oct 2022

TL;DR: In this article , the authors generalize the result of McDonald and Taylor to compact sets in R d and show that the edge lengths in C 1 × C 2 corresponding to any pinned finite tree configuration have non-empty interior.

Abstract: A BSTRACT . We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set C 1 × C 2 utilizing the notion of thickness. Speciﬁcally, we show that C 1 , C 2 ⊂ R d compact sets with thickness satisfying τ ( C 1 ) τ ( C 2 ) > 1 , then the edge lengths in C 1 × C 2 corresponding to any pinned ﬁnite tree conﬁguration has non-empty interior. Originally proven for Cantor sets on the real line by McDonald and Taylor, we use the notion of thickness introduced by Falconer and Yavicoli which allows us to generalize the result of McDonald and Taylor to compact sets in R d .

1 citations

29 May 2023

TL;DR: In this paper , restricted Falconer distance problems are introduced, which lie in between the classical version and its pinned variant, and they are shown to have non-empty interior if the diagonal distance set has a nonempty interior.

Abstract: We introduce a new class of Falconer distance problems, which we call restricted Falconer distance problems, that lie in between the classical version and its pinned variant. A particular model we study is the diagonal distance set $$\Delta^{diag}(E)= \{ \,|(x,x)-(y_1,y_2)| \, :\, x,\,y_1,\,y_2\, \in E\, \}$$ which we show has non-empty interior if $\dim(E)>\frac{2d+1}{3}$. Standard pinned variants of the Falconer distance problem either can not guarantee a pin on the diagonal or yield worse dimensional thresholds. A key tool for our result is an $L^p$ improving estimate for the bilinear spherical averaging operator with decay on frequency scales.

1 citations