arXiv subjects

Livia Betti

Publications and source records attributed to Livia Betti.

7 recordsLinked to original sources

arXiv2026

What's in an Earth Embedding? An Explainability Analysis of Location Encoders

Geographic implicit neural representations (INRs) learn to map any coordinate on Earth to a location embedding, implicitly encoding geospatial data into the weights of a neural network. Location embeddings are widely used off the shelf as general-purpose geospatial representations, yet users lack principled tools to audit what geographic or semantic information these embeddings capture. In this work, we analyze the information content of geographic INRs through their location embeddings. We decompose these embeddings into human-interpretable features$\unicode{x2014}$namely, (i) sparse latent concepts, (ii) natural language concepts, and (iii) visual features. The latent concept embeddings are learned using sparse autoencoders. To recover natural language concepts, we apply sparse linear concept embeddings (SpLiCE) over a predefined geospatial dictionary. Finally, visual features are extracted using saliency maps derived from CLIP Surgery. We show that location embeddings can be decomposed into human-interpretable representations while retaining high reconstruction capability, revealing interpretable geographic structures such as forests, deserts, and urban features. Across methods, sparse decompositions expose systematic differences in encoded information, ranging from urban structures to broader biome and climate signals, and pretraining-space saliency maps further highlight complementary features such as roads and landmarks. We hope this work provides a first step toward interpretable geospatial representations.

cs.LG↗

arXiv2026

OT on the Map: Quantifying Domain Shifts in Geographic Space

In computer vision and machine learning for geographic data, out-of-domain generalization is a pervasive challenge, arising from uneven global data coverage and distribution shifts across geographic regions. Though models are frequently trained in one region and deployed in another, there is no principled method for determining when this cross-region adaptation will be successful. A well-defined notion of distance between distributions can effectively quantify how different a new target domain is compared to the domains used for model training, which in turn could support model training and deployment decisions. In this paper, we propose a strategy for computing distances between geospatial domains that leverages geographic information with Optimal Transport methods (GeoSpOT). In our experiments, GeoSpOT distances emerge as effective predictors of cross-domain transfer difficulty. We further demonstrate that embeddings from pretrained location encoders provide information comparable to image/text embeddings, despite relying solely on longitude-latitude pairs as input. This allows users to get an approximation of out-of-domain performance for geospatial models, even when the exact downstream task is unknown, or no task-specific data is available. Building on these findings, we show that GeoSpOT distances can preemptively guide data selection and enable predictive tools to analyze regions where a model is likely to underperform.

cs.LG↗

arXiv2025

Mapping on a Budget: Optimizing Spatial Data Collection for ML

In applications across agriculture, ecology, and human development, machine learning with satellite imagery (SatML) is limited by the sparsity of labeled training data. While satellite data cover the globe, labeled training datasets for SatML are often small, spatially clustered, and collected for other purposes (e.g., administrative surveys or field measurements). Despite the pervasiveness of this issue in practice, past SatML research has largely focused on new model architectures and training algorithms to handle scarce training data, rather than modeling data conditions directly. This leaves scientists and policymakers who wish to use SatML for large-scale monitoring uncertain about whether and how to collect additional data to maximize performance. Here, we present the first problem formulation for the optimization of spatial training data in the presence of heterogeneous data collection costs and realistic budget constraints, as well as novel methods for addressing this problem. In experiments simulating different problem settings across three continents and four tasks, our strategies reveal substantial gains from sample optimization. Further experiments delineate settings for which optimized sampling is particularly effective. The problem formulation and methods we introduce are designed to generalize across application domains for SatML; we put special emphasis on a specific problem setting where our coauthors can immediately use our findings to augment clustered agricultural surveys for SatML monitoring in Togo.

cs.LG↗

arXiv2023

VC-Dimension of Hyperplanes over Finite Fields

Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$.

math.CO↗

arXiv2023

Benfordness of Measurements Resulting from Box Fragmentation

We make progress on a conjecture made by [DM], which states that the $d$-dimensional frames of $m$-dimensional boxes resulting from a fragmentation process satisfy Benford's law for all $1 \leq d \leq m$. We provide a sufficient condition for Benford's law to be satisfied, namely that the maximum product of $d$ sides is itself a Benford random variable. Motivated to produce an example of such a fragmentation process, we show that processes constructed from log-uniform proportion cuts satisfy the maximum criterion for $d=1$.

math.PR↗

arXiv2022

VC-Dimension and Distance Chains in $\mathbb{F}_q^d$

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 $|E|\geq C_2 q^{7/4}$.

math.CO↗

arXiv2022

Distinct Angles and Angle Chains in Three Dimensions

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 $\mathbb{R}^3$.

cs.CG↗