arXiv subjects

Triet M. Le

Publications and source records attributed to Triet M. Le.

5 recordsLinked to original sources

arXiv2026

Square Functions and Rectifiability under Monotone Transformations of the Density

Let $μ$ be an $n$-AD-regular measure in $\mathbb{R}^d$. Chousionis, Garnett, Le and Tolsa [CGLT] proved that $μ$ is uniformly $n$-rectifiable if and only if the square function built from the density differences $Δ_μ(x,r)=μ(B(x,r))/r^n-μ(B(x,2r))/(2r)^n$ satisfies a Carleson condition. In this paper we show that the same characterization holds if the density is first composed with a function $F$ which is bi-Lipschitz on the interval $[c_0^{-1},c_0]$ determined by the AD-regularity constant $c_0$. The main example is $F=\log$, introduced in [Le], for which the square function takes the scale-invariant form $Δ_μ^{\log}(x,r) = \log\bigl(μ(B(x,r))/μ(B(x,2r))\bigr)+n\log 2$. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of $μ$ with a Gaussian or a more general radial kernel, discuss what happens when $F$ is not bi-Lipschitz, and treat the case $μ(\mathbb{R}^d)<\infty$, where the behavior of $F$ near zero enters in only one of the two implications. We also show that the qualitative characterization of $n$-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at $μ$-almost every point, holds after composition with any locally bi-Lipschitz $F$. This requires neither AD-regularity nor doubling, and for $F=\log$ the condition $\lim_{r\to0}Δ_μ(x,r)=0$ becomes $\lim_{r\to0}μ(B(x,r))/μ(B(x,2r))=2^{-n}$.

math.CA↗

arXiv2026

UR-JEPA: Uniform Rectifiability as a Regularizer for Joint-Embedding Predictive Architectures

A central difficulty in training Joint-Embedding Predictive Architectures (JEPAs) is preventing representation collapse. LeJEPA addresses this by enforcing an isotropic Gaussian target on the embeddings via Sketched Isotropic Gaussian Regularization (SIGReg). This target is in tension with the manifold hypothesis, which expects embeddings to concentrate on a low-dimensional subset of the ambient space. We propose \emph{UR-JEPA}, which targets a uniformly $n$-rectifiable measure of local tangent dimension $n$ at small scales, realized through a Gaussian-kernel smoothed Carleson-type square function $\mathcal{L}^{\text{CGLT}}$, with a complementary Jones $β$-number formulation. On Inet10, UR-JEPA($\mathcal{L}^{\text{CGLT}}$) attains $0.9141 \pm 0.0014$ for a $+0.83$\,pp gain over LeJEPA($\mathcal{L}^{\text{SIGReg}}$) with $\sim 30\%$ lower seed standard deviation; on matched-recipe Galaxy10~SDSS, a single-seed ImageNet-$100$ run, and a $3$-seed EuroSAT remote-sensing run, the two methods lie in the same peak-accuracy band at convergence, with UR-JEPA retaining its lower-seed-variance signature. On EuroSAT the in-domain pair is competitive at $96.0$ to $96.1\%$ with large remote-sensing foundation-model transfer at a $25\times$ smaller backbone. The distinction is geometric: direct visualization of the projector output distribution shows that on all four datasets UR--JEPA($\mathcal{L}^{\text{CGLT}}$) produces a global PCA spectrum with a $4$ to $5$ order-of-magnitude drop at index $\sim 20$ to $25$ out of $D = 32$, while LeJEPA's spectrum is near-flat (top-to-bottom ratio at most $3.6$). Per-dimension marginals are simultaneously near-Gaussian for both methods (mean Shapiro-Wilk $W \in [0.992, 0.996]$) as a Diaconis-Freedman consequence. At matched accuracy the two regularizers therefore yield structurally distinct projected representations.

cs.LG↗

arXiv2025

BIG5-TPoT: Predicting BIG Five Personality Traits, Facets, and Items Through Targeted Preselection of Texts

Predicting an individual's personalities from their generated texts is a challenging task, especially when the text volume is large. In this paper, we introduce a straightforward yet effective novel strategy called targeted preselection of texts (TPoT). This method semantically filters the texts as input to a deep learning model, specifically designed to predict a Big Five personality trait, facet, or item, referred to as the BIG5-TPoT model. By selecting texts that are semantically relevant to a particular trait, facet, or item, this strategy not only addresses the issue of input text limits in large language models but also improves the Mean Absolute Error and accuracy metrics in predictions for the Stream of Consciousness Essays dataset.

cs.CL↗

arXiv2019

A Robust Time Series Model with Outliers and Missing Entries

This paper studies the problem of robustly learning the correlation function for a univariate time series with the presence of noise, outliers and missing entries. The outliers or anomalies considered here are sparse and rare events that deviate from normality which is depicted by a correlation function and an uncertainty condition. This general formulation is applied to univariate time series of event counts (or non-negative time series) where the correlation is a log-linear function with the uncertainty condition following the Poisson distribution. Approximations to the sparsity constraint, such as $\ell^r, 0< r\le 1$, are used to obtain robustness in the presence of outliers. The $\ell^r$ constraint is also applied to the correlation function to reduce the number of active coefficients. This task also helps bypassing the model selection procedure. Simulated results are presented to validate the model.

stat.AP↗

arXiv2012

Image Processing Variations with Analytic Kernels

Let $f\in L^1(\R^d)$ be real. The Rudin-Osher-Fatemi model is to minimize $\|u\|_{\dot{BV}}+λ\|f-u\|_{L^2}^2$, in which one thinks of $f$ as a given image, $λ> 0$ as a "tuning parameter", $u$ as an optimal "cartoon" approximation to $f$, and $f-u$ as "noise" or "texture". Here we study variations of the R-O-F model having the form $\inf_u\{\|u\|_{\dot{BV}}+λ\|K*(f-u)\|_{L^p}^q\}$ where $K$ is a real analytic kernel such as a Gaussian. For these functionals we characterize the minimizers $u$ and establish several of their properties, including especially their smoothness properties. In particular we prove that on any open set on which $u \in W^{1,1}$ and $\nabla u \neq 0$ almost every level set $\{u =c\}$ is a real analytic surface. We also prove that if $f$ and $K$ are radial functions then every minimizer $u$ is a radial step function.

math.AP↗