arXiv subjects

Elvira Mayordomo

Publications and source records attributed to Elvira Mayordomo.

At least 19 recordsLinked to original sources

arXiv2026

Spatial Normalization for Cross-Domain Retinal Layer Segmentation in Optical Coherence Tomography

Retinal layer segmentation in Optical Coherence Tomography (OCT) is a fundamental step for extracting quantitative biomarkers of retinal structure. Indeed, there is a growing interest in the analysis of OCTs in the context of neurodegenerative diseases. However, segmentation remains challenging due to speckle noise, shadowing artifacts, low contrast between adjacent layers, anatomical variability across subjects, and domain shifts arising from different acquisition protocols and clinical populations. While deep learning methods have achieved remarkable performance, their robustness and generalization across heterogeneous datasets remain limited. In this work, we investigate the role of spatial normalization as a preprocessing strategy to mitigate geometric domain shifts and improve the consistency of retinal layer segmentation. Inspired by standard practices in neuroimaging, we introduce a fovea-centered normalization framework that aligns OCT volumes into a common anatomical reference. We perform a comprehensive evaluation of state-of-the-art deep learning architectures. To provide a comprehensive assessment of segmentation quality, we combine conventional overlap-based metrics at B-scan level with topology-aware metrics at A-scan level and thickness-based measures at the en-face level. In cases where a ground truth is not available, we propose topology violation quantitative metrics that do not require ground truth annotations and a thickness-based qualitative assessment that captures structural consistency and clinically relevant patterns at the en-face level. The results demonstrate the importance of spatial normalization in OCT segmentation pipelines toward the development of robust and clinically meaningful retinal analysis tools, enabling reliable biomarker extraction and downstream computational analysis in neurodegenerative research.

cs.CV↗

arXiv2026

Algorithmic Information Bounds for Distances and Orthogonal Projections

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

cs.CC↗

arXiv2026

Fractal dimensions and profinite groups

Let $T$ be a finitely branching rooted tree such that any node has at least two successors. The path space $[T]$ is an ultrametric space: for distinct paths $f,g$ let $d(f,g)= 1/|T_n|$, where $T_n$ denotes the $n$-th level of the tree, and $n$ is largest such that $f(n)= g(n)$. Let $S$ be a subtree of $T$ without leaves that is level-wise uniformly branching, in the sense that the number of successors of a node only depends on its level. We~show that the Hausdorff and lower box dimensions coincide for~$[S]$, and the packing and upper box dimensions also coincide. We give geometric proofs, as well as proofs based on the point-to-set principles. We use the first result to reprove a theorem of Barnea and Shalev on the Hausdorff dimension of closed subgroups of a profinite group $G$, referring only on the geometric structure of the closed subgroup in the canonical path space given by an inverse system for $G$. We obtain an analogous theorem for the packing dimension.

math.GR↗

arXiv2025

Bounding the dimension of exceptional sets for orthogonal projections

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(π_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $π_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(π_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for $k=1$ and for $k=n-1$. Hence we completely resolve this question for $n=3$.

math.CA↗

arXiv2025

Normality, Relativization, and Randomness

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of supernormal numbers that correspond to semicomputable relativizations, and that of highly normal numbers in terms of computable ones. We prove several properties of these new randomness concepts. Both supernormality and high normality generalize Borel absolute normality. Supernormality is strictly between 2-randomness and effective dimension 1, while high normality corresponds exactly to sequences of computable dimension 1 providing a more natural characterization of this class.

math.LO↗

arXiv2025

A point to set principle for finite-state dimension

Effective dimension has proven very useful in geometric measure theory through the point-to-set principle \cite{LuLu18}\ that characterizes Hausdorff dimension by relativized effective dimension. Finite-state dimension is the least demanding effectivization in this context \cite{FSD}\ that among other results can be used to characterize Borel normality \cite{BoHiVi05}. In this paper we prove a characterization of finite-state dimension in terms of information content of a real number at a certain precision. We then use this characterization to give a robust concept of relativized normality and prove a finite-state dimension point-to-set principle. We finish with an open question on the equidistribution properties of relativized normality.

cs.CC↗

arXiv2023

Explainable artificial intelligence toward usable and trustworthy computer-aided early diagnosis of multiple sclerosis from Optical Coherence Tomography

Background: Several studies indicate that the anterior visual pathway provides information about the dynamics of axonal degeneration in Multiple Sclerosis (MS). Current research in the field is focused on the quest for the most discriminative features among patients and controls and the development of machine learning models that yield computer-aided solutions widely usable in clinical practice. However, most studies are conducted with small samples and the models are used as black boxes. Clinicians should not trust machine learning decisions unless they come with comprehensive and easily understandable explanations. Materials and methods: A total of 216 eyes from 111 healthy controls and 100 eyes from 59 patients with relapsing-remitting MS were enrolled. The feature set was obtained from the thickness of the ganglion cell layer (GCL) and the retinal nerve fiber layer (RNFL). Measurements were acquired by the novel Posterior Pole protocol from Spectralis Optical Coherence Tomography (OCT) device. We compared two black-box methods (gradient boosting and random forests) with a glass-box method (explainable boosting machine). Explainability was studied using SHAP for the black-box methods and the scores of the glass-box method. Results: The best-performing models were obtained for the GCL layer. Explainability pointed out to the temporal location of the GCL layer that is usually broken or thinning in MS and the relationship between low thickness values and high probability of MS, which is coherent with clinical knowledge. Conclusions: The insights on how to use explainability shown in this work represent a first important step toward a trustworthy computer-aided solution for the diagnosis of MS with OCT.

eess.IV↗

arXiv2021

LDDMM meets GANs: Generative Adversarial Networks for diffeomorphic registration

The purpose of this work is to contribute to the state of the art of deep-learning methods for diffeomorphic registration. We propose an adversarial learning LDDMM method for pairs of 3D mono-modal images based on Generative Adversarial Networks. The method is inspired by the recent literature for deformable image registration with adversarial learning. We combine the best performing generative, discriminative, and adversarial ingredients from the state of the art within the LDDMM paradigm. We have successfully implemented two models with the stationary and the EPDiff-constrained non-stationary parameterizations of diffeomorphisms. Our unsupervised and data-hungry approach has shown a competitive performance with respect to a benchmark supervised and rich-data approach. In addition, our method has shown similar results to model-based methods with a computational time under one second.

cs.CV↗

arXiv2021

Dimension and the Structure of Complexity Classes

We prove three results on the dimension structure of complexity classes. 1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set $X$ of languages in terms of the relativized resource-bounded dimensions of the individual elements of $X$, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class $X$ of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of $X$. 2. Every language that is $\leq^P_m$-reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is $\leq^P_m$-hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.

cs.CC↗

arXiv2021

Extending the Reach of the Point-to-Set Principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces $X$. We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to individual points $x\in X$--to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages $E_0, E_1, E_2, \ldots$ used to construct a self-similar fractal $E$ in the plane are elements of the hyperspace of the plane, and they converge to $E$ in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets $E$ that are analytic, i.e., $\mathbfΣ^1_1$, a tight bound on the packing dimension of the hyperspace of $E$ in terms of the packing dimension of $E$ itself.

cs.CC↗

arXiv2020

Algorithmic Fractal Dimensions in Geometric Measure Theory

The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections with computable functions on the reals, recent uses of algorithmic dimensions in proving new theorems in classical (non-algorithmic) fractal geometry, and directions for future research.

cs.CC↗

arXiv2020

Computing Absolutely Normal Numbers in Nearly Linear Time

A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number $x$, with the $n$th bit of $x$ appearing after $n$polylog$(n)$ computation steps. This speed is achieved by simultaneously computing and diagonalizing against a martingale that incorporates Lempel-Ziv parsing algorithms in all bases.

cs.DS↗

arXiv2019

Asymptotic Divergences and Strong Dichotomy

The Schnorr-Stimm dichotomy theorem concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet $Σ$. In this paper we use the Kullback-Leibler divergence to formulate the $\textit{lower asymptotic divergence}$ $\text{div}(S||α)$ of a probability measure $α$ on $Σ$ from a sequence $S$ over $Σ$ and the $\textit{upper asymptotic divergence}$ $\text{Div}(S||α)$ of $α$ from $S$ in such a way that a sequence $S$ is $α$-normal (meaning that every string $w$ has asymptotic frequency $α(w)$ in $S$) if and only if $\text{Div}(S||α)=0$. We also use the Kullback-Leibler divergence to quantify the $\textit{total risk }$ $\text{Risk}_G(w)$ that a finite-state gambler $G$ takes when betting along a prefix $w$ of $S$. Our main theorem is a $\textit{strong dichotomy theorem}$ that uses the above notions to $\textit{quantify}$ the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to $α$-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes $w$ of $S$. (1) The infinitely-often exponential rate of winning is $2^{\text{Div}(S||α)|w|}$. (2) The exponential rate of loss is $2^{-\text{Risk}_G(w)}$. We also use (1) to show that $1-\text{Div}(S||α)/c$, where $c= \log(1/ \min_{a\inΣ}α(a))$, is an upper bound on the finite-state $α$-dimension of $S$ and prove the dual fact that $1-\text{div}(S||α)/c$ is an upper bound on the finite-state strong $α$-dimension of $S$.

cs.IT↗

arXiv2017

Effective dimension in some general metric spaces

We introduce the concept of effective dimension for a wide class of metric spaces that are not required to have a computable measure. Effective dimension was defined by Lutz in (Lutz 2003) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated.

cs.CC↗

arXiv2011

Resource-bounded Dimension in Computational Learning Theory

This paper focuses on the relation between computational learning theory and resource-bounded dimension. We intend to establish close connections between the learnability/nonlearnability of a concept class and its corresponding size in terms of effective dimension, which will allow the use of powerful dimension techniques in computational learning and viceversa, the import of learning results into complexity via dimension. Firstly, we obtain a tight result on the dimension of online mistake-bound learnable classes. Secondly, in relation with PAC learning, we show that the polynomial-space dimension of PAC learnable classes of concepts is zero. This provides a hypothesis on effective dimension that implies the inherent unpredictability of concept classes (the classes that verify this property are classes not efficiently PAC learnable using any hypothesis). Thirdly, in relation to space dimension of classes that are learnable by membership query algorithms, the main result proves that polynomial-space dimension of concept classes learnable by a membership-query algorithm is zero.

cs.CC↗

arXiv2010

Inseparability and Strong Hypotheses for Disjoint NP Pairs

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k))-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs.

cs.CC↗

arXiv2009

Curves That Must Be Retraced

We exhibit a polynomial time computable plane curve GAMMA that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization f of GAMMA and every positive integer n, there is some positive-length subcurve of GAMMA that f retraces at least n times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.

cs.CC↗

arXiv2009

Polylog space compression, pushdown compression, and Lempel-Ziv are incomparable

The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation, and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose Lempel-Ziv algorithm. This comparison is made without any a priori assumption on the data's source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.

cs.CC↗