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Buch ; Online: Sampling, Diffusions, and Stochastic Localization

Montanari, Andrea

2023

Abstract: Diffusions are a successful technique to sample from high-dimensional distributions can be either explicitly given or learnt from a collection of samples. They implement a diffusion process whose endpoint is a sample from the target distribution and ... ...

Abstract	Diffusions are a successful technique to sample from high-dimensional distributions can be either explicitly given or learnt from a collection of samples. They implement a diffusion process whose endpoint is a sample from the target distribution and whose drift is typically represented as a neural network. Stochastic localization is a successful technique to prove mixing of Markov Chains and other functional inequalities in high dimension. An algorithmic version of stochastic localization was introduced in [EAMS2022], to obtain an algorithm that samples from certain statistical mechanics models. This notes have three objectives: (i) Generalize the construction [EAMS2022] to other stochastic localization processes; (ii) Clarify the connection between diffusions and stochastic localization. In particular we show that standard denoising diffusions are stochastic localizations but other examples that are naturally suggested by the proposed viewpoint; (iii) Describe some insights that follow from this viewpoint. Comment: 31 pages, 5 pdf figures
Schlagwörter	Computer Science - Machine Learning
Thema/Rubrik (Code)	519
Erscheinungsdatum	2023-05-18
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Artikel ; Online: Urban environment influences on stress, autonomic reactivity and circadian rhythm: protocol for an ambulatory study of mental health and sleep.

Montanari, Andrea / Wang, Limin / Birenboim, Amit / Chaix, Basile

Frontiers in public health

2024 Band 12, Seite(n) 1175109

Abstract: Introduction: Converging evidence suggests that urban living is associated with an increased likelihood of developing mental health and sleep problems. Although these aspects have been investigated in separate streams of research, stress, autonomic ... ...

Abstract	Introduction: Converging evidence suggests that urban living is associated with an increased likelihood of developing mental health and sleep problems. Although these aspects have been investigated in separate streams of research, stress, autonomic reactivity and circadian misalignment can be hypothesized to play a prominent role in the causal pathways underlining the complex relationship between the urban environment and these two health dimensions. This study aims at quantifying the momentary impact of environmental stressors on increased autonomic reactivity and circadian rhythm, and thereby on mood and anxiety symptoms and sleep quality in the context of everyday urban living. Method: The present article reports the protocol for a feasibility study that aims at assessing the daily environmental and mobility exposures of 40 participants from the urban area of Jerusalem over 7 days. Every participant will carry a set of wearable sensors while being tracked through space and time with GPS receivers. Skin conductance and heart rate variability will be tracked to monitor participants' stress responses and autonomic reactivity, whereas electroencephalographic signal will be used for sleep quality tracking. Light exposure, actigraphy and skin temperature will be used for ambulatory circadian monitoring. Geographically explicit ecological momentary assessment (GEMA) will be used to assess participants' perception of the environment, mood and anxiety symptoms, sleep quality and vitality. For each outcome variable (sleep quality and mental health), hierarchical mixed models including random effects at the individual level will be used. In a separate analysis, to control for potential unobserved individual-level confounders, a fixed effect at the individual level will be specified for case-crossover analyses (comparing each participant to oneself). Conclusion: Recent developments in wearable sensing methods, as employed in our study or with even more advanced methods reviewed in the Discussion, make it possible to gather information on the functioning of neuro-endocrine and circadian systems in a real-world context as a way to investigate the complex interactions between environmental exposures, behavior and health. Our work aims to provide evidence on the health effects of urban stressors and circadian disruptors to inspire potential interventions, municipal policies and urban planning schemes aimed at addressing those factors.
Mesh-Begriff(e)	Humans ; Mental Health ; Sleep/physiology ; Circadian Rhythm/physiology ; Actigraphy ; Affect
Sprache	Englisch
Erscheinungsdatum	2024-02-05
Erscheinungsland	Switzerland
Dokumenttyp	Journal Article ; Research Support, Non-U.S. Gov't
ZDB-ID	2711781-9
ISSN	2296-2565 ; 2296-2565
ISSN (online)	2296-2565
ISSN	2296-2565
DOI	10.3389/fpubh.2024.1175109
Datenquelle	MEDical Literature Analysis and Retrieval System OnLINE

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Buch ; Online: Compressing Tabular Data via Latent Variable Estimation

Montanari, Andrea / Weiner, Eric

2023

Abstract: Data used for analytics and machine learning often take the form of tables with categorical entries. We introduce a family of lossless compression algorithms for such data that proceed in four steps: $(i)$ Estimate latent variables associated to rows and ...

Abstract	Data used for analytics and machine learning often take the form of tables with categorical entries. We introduce a family of lossless compression algorithms for such data that proceed in four steps: $(i)$ Estimate latent variables associated to rows and columns; $(ii)$ Partition the table in blocks according to the row/column latents; $(iii)$ Apply a sequential (e.g. Lempel-Ziv) coder to each of the blocks; $(iv)$ Append a compressed encoding of the latents. We evaluate it on several benchmark datasets, and study optimal compression in a probabilistic model for that tabular data, whereby latent values are independent and table entries are conditionally independent given the latent values. We prove that the model has a well defined entropy rate and satisfies an asymptotic equipartition property. We also prove that classical compression schemes such as Lempel-Ziv and finite-state encoders do not achieve this rate. On the other hand, the latent estimation strategy outlined above achieves the optimal rate. Comment: 45 pages; 6 pdf figures
Schlagwörter	Computer Science - Information Theory
Thema/Rubrik (Code)	006
Erscheinungsdatum	2023-02-20
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Buch ; Online: Adversarial Examples in Random Neural Networks with General Activations

Montanari, Andrea / Wu, Yuchen

2022

Abstract: A substantial body of empirical work documents the lack of robustness in deep learning models to adversarial examples. Recent theoretical work proved that adversarial examples are ubiquitous in two-layers networks with sub-exponential width and ReLU or ... ...

Abstract	A substantial body of empirical work documents the lack of robustness in deep learning models to adversarial examples. Recent theoretical work proved that adversarial examples are ubiquitous in two-layers networks with sub-exponential width and ReLU or smooth activations, and multi-layer ReLU networks with sub-exponential width. We present a result of the same type, with no restriction on width and for general locally Lipschitz continuous activations. More precisely, given a neural network $f(\,\cdot\,;{\boldsymbol \theta})$ with random weights ${\boldsymbol \theta}$, and feature vector ${\boldsymbol x}$, we show that an adversarial example ${\boldsymbol x}'$ can be found with high probability along the direction of the gradient $\nabla_{{\boldsymbol x}}f({\boldsymbol x};{\boldsymbol \theta})$. Our proof is based on a Gaussian conditioning technique. Instead of proving that $f$ is approximately linear in a neighborhood of ${\boldsymbol x}$, we characterize the joint distribution of $f({\boldsymbol x};{\boldsymbol \theta})$ and $f({\boldsymbol x}';{\boldsymbol \theta})$ for ${\boldsymbol x}' = {\boldsymbol x}-s({\boldsymbol x})\nabla_{{\boldsymbol x}}f({\boldsymbol x};{\boldsymbol \theta})$. Comment: 36 pages
Schlagwörter	Computer Science - Machine Learning ; Computer Science - Cryptography and Security ; Mathematics - Statistics Theory
Thema/Rubrik (Code)	006
Erscheinungsdatum	2022-03-31
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Buch ; Online: Universality of empirical risk minimization

Montanari, Andrea / Saeed, Basil

2022

Abstract: Consider supervised learning from i.i.d. samples $\{{\boldsymbol x}_i,y_i\}_{i\le n}$ where ${\boldsymbol x}_i \in\mathbb{R}^p$ are feature vectors and ${y} \in \mathbb{R}$ are labels. We study empirical risk minimization over a class of functions that ... ...

Abstract	Consider supervised learning from i.i.d. samples $\{{\boldsymbol x}_i,y_i\}_{i\le n}$ where ${\boldsymbol x}_i \in\mathbb{R}^p$ are feature vectors and ${y} \in \mathbb{R}$ are labels. We study empirical risk minimization over a class of functions that are parameterized by $\mathsf{k} = O(1)$ vectors ${\boldsymbol \theta}_1, . . . , {\boldsymbol \theta}_{\mathsf k} \in \mathbb{R}^p$ , and prove universality results both for the training and test error. Namely, under the proportional asymptotics $n,p\to\infty$, with $n/p = \Theta(1)$, we prove that the training error depends on the random features distribution only through its covariance structure. Further, we prove that the minimum test error over near-empirical risk minimizers enjoys similar universality properties. In particular, the asymptotics of these quantities can be computed $-$to leading order$-$ under a simpler model in which the feature vectors ${\boldsymbol x}_i$ are replaced by Gaussian vectors ${\boldsymbol g}_i$ with the same covariance. Earlier universality results were limited to strongly convex learning procedures, or to feature vectors ${\boldsymbol x}_i$ with independent entries. Our results do not make any of these assumptions. Our assumptions are general enough to include feature vectors ${\boldsymbol x}_i$ that are produced by randomized featurization maps. In particular we explicitly check the assumptions for certain random features models (computing the output of a one-layer neural network with random weights) and neural tangent models (first-order Taylor approximation of two-layer networks). Comment: 74 pages
Schlagwörter	Mathematics - Statistics Theory ; Computer Science - Machine Learning ; Statistics - Machine Learning
Thema/Rubrik (Code)	519
Erscheinungsdatum	2022-02-17
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Buch ; Online: Overparametrized linear dimensionality reductions

Montanari, Andrea / Zhou, Kangjie

From projection pursuit to two-layer neural networks

2022

Abstract: Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability ... ...

Abstract	Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large? We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\to\alpha\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, \alpha}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, \alpha}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,\alpha}$ up to logarithmic factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and R\'{e}nyi information dimension. The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons. Comment: 53 pages, 1 figure, an earlier version of this paper was accepted for presentation at the Conference on Learning Theory (COLT) 2022
Schlagwörter	Statistics - Machine Learning ; Computer Science - Machine Learning
Thema/Rubrik (Code)	519
Erscheinungsdatum	2022-06-13
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Artikel ; Online: Nonnegative Matrix Factorization Via Archetypal Analysis

Javadi, Hamid / Montanari, Andrea

Journal of the American Statistical Association. 2020 Apr. 2, v. 115, no. 530 p.896-907

2020

Abstract: Given a collection of data points, nonnegative matrix factorization (NMF) suggests expressing them as convex combinations of a small set of “archetypes” with nonnegative entries. This decomposition is unique only if the true archetypes are nonnegative ... ...

Abstract	Given a collection of data points, nonnegative matrix factorization (NMF) suggests expressing them as convex combinations of a small set of “archetypes” with nonnegative entries. This decomposition is unique only if the true archetypes are nonnegative and sufficiently sparse (or the weights are sufficiently sparse), a regime that is captured by the separability condition and its generalizations. In this article, we study an approach to NMF that can be traced back to the work of Cutler and Breiman [(1994), “Archetypal Analysis,” Technometrics, 36, 338–347] and does not require the data to be separable, while providing a generally unique decomposition. We optimize a trade-off between two objectives: we minimize the distance of the data points from the convex envelope of the archetypes (which can be interpreted as an empirical risk), while also minimizing the distance of the archetypes from the convex envelope of the data (which can be interpreted as a data-dependent regularization). The archetypal analysis method of Cutler and Breiman is recovered as the limiting case in which the last term is given infinite weight. We introduce a “uniqueness condition” on the data which is necessary for identifiability. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. While our approach requires solving a nonconvex optimization problem, we find that standard optimization methods succeed in finding good solutions for both real and synthetic data. Supplementary materials for this article are available online
Schlagwörter	algorithms ; data collection ; geometry ; risk ; system optimization ; Dimensionality reduction ; Matrix factorization ; Separability
Sprache	Englisch
Erscheinungsverlauf	2020-0402
Umfang	p. 896-907.
Erscheinungsort	Taylor & Francis
Dokumenttyp	Artikel ; Online
ZDB-ID	2064981-2
ISSN	1537-274X
ISSN	1537-274X
DOI	10.1080/01621459.2019.1594832
Datenquelle	NAL Katalog (AGRICOLA)

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Buch ; Online: Learning time-scales in two-layers neural networks

Berthier, Raphaël / Montanari, Andrea / Zhou, Kangjie

2023

Abstract: Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely any ... ...

Abstract	Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely any progress alternate with intervals of rapid decrease. These successive phases of learning often take place on very different time scales. Finally, models learnt in an early phase are typically `simpler' or `easier to learn' although in a way that is difficult to formalize. Although theoretical explanations of these phenomena have been put forward, each of them captures at best certain specific regimes. In this paper, we study the gradient flow dynamics of a wide two-layer neural network in high-dimension, when data are distributed according to a single-index model (i.e., the target function depends on a one-dimensional projection of the covariates). Based on a mixture of new rigorous results, non-rigorous mathematical derivations, and numerical simulations, we propose a scenario for the learning dynamics in this setting. In particular, the proposed evolution exhibits separation of timescales and intermittency. These behaviors arise naturally because the population gradient flow can be recast as a singularly perturbed dynamical system. Comment: 54 pages, 9 figures
Schlagwörter	Computer Science - Machine Learning ; Mathematics - Optimization and Control ; Statistics - Machine Learning
Thema/Rubrik (Code)	006
Erscheinungsdatum	2023-02-28
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Buch ; Online: Towards a statistical theory of data selection under weak supervision

Kolossov, Germain / Montanari, Andrea / Tandon, Pulkit

2023

Abstract: Given a sample of size $N$, it is often useful to select a subsample of smaller size $n

Abstract	Given a sample of size $N$, it is often useful to select a subsample of smaller size $n<N$ to be used for statistical estimation or learning. Such a data selection step is useful to reduce the requirements of data labeling and the computational complexity of learning. We assume to be given $N$ unlabeled samples $\{{\boldsymbol x}_i\}_{i\le N}$, and to be given access to a `surrogate model' that can predict labels $y_i$ better than random guessing. Our goal is to select a subset of the samples, to be denoted by $\{{\boldsymbol x}_i\}_{i\in G}$, of size $\|G\|=n<N$. We then acquire labels for this set and we use them to train a model via regularized empirical risk minimization. By using a mixture of numerical experiments on real and synthetic data, and mathematical derivations under low- and high- dimensional asymptotics, we show that: $(i)$~Data selection can be very effective, in particular beating training on the full sample in some cases; $(ii)$~Certain popular choices in data selection methods (e.g. unbiased reweighted subsampling, or influence function-based subsampling) can be substantially suboptimal.<br /> Comment: 55 pages; 14 figures
Schlagwörter	Statistics - Machine Learning ; Computer Science - Machine Learning
Thema/Rubrik (Code)	310
Erscheinungsdatum	2023-09-25
Erscheinungsland	us
Dokumenttyp	Buch ; Online
Datenquelle	BASE - Bielefeld Academic Search Engine (Lebenswissenschaftliche Auswahl)

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Artikel: The Slinky King: Western Attitudes toward the Durlan in Colonial Southeast Asia

Montanari, Andrea

Food, culture & society

2017 Band 20, Heft 3, Seite(n) 395

Sprache	Englisch
Dokumenttyp	Artikel
ZDB-ID	2240542-2
ISSN	1552-8014
Datenquelle	Current Contents Ernährung, Umwelt, Agrar

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