Search

High-content omic technologies coupled with sparsity-promoting regularization methods (SRM) have transformed the biomarker discovery process. However, the translation of computational results into a clinical use-case scenario remains challenging. A rate-limiting step is the rigorous selection of reliable biomarker candidates among a host of biological features included in multivariate models. We propose Stabl, a machine learning framework that unifies the biomarker discovery process with multivariate predictive modeling of clinical outcomes by selecting a sparse and reliable set of biomarkers. Evaluation of Stabl on synthetic datasets and four independent clinical studies demonstrates improved biomarker sparsity and reliability compared to commonly used SRMs at similar predictive performance. Stabl readily extends to double- and triple-omics integration tasks and identifies a sparser and more reliable set of biomarkers than those selected by state-of-the-art early- and late-fusion SRMs, thereby facilitating the biological interpretation and clinical translation of complex multi-omic predictive models. The complete package for Stabl is available online at https://github.com/gregbellan/Stabl.

L’articolo racconta, a partire dal film del 2018 Stan & Ollie, il viaggio in Italia del duo comico più famoso del mondo, datato 1950. Fra il 1947 e il 1954, quando il successo in America iniziava a scemare, Stanlio e Ollio intrapresero una tournée europea in Francia, Inghilterra, Belgio e Danimarca, accolti ovunque in trionfo. Qui, infatti, il loro successo sembrava non aver mai avuto un calo; ovunque andassero, la risposta del pubblico era sempre la stessa: una grande e calorosa accoglienza.

L'associazionismo giovanile di partito č una realtŕ a tutt'oggi indagata solo in parte dalla ricerca storica. Il saggio ricostruisce, tramite una pluralitŕ di fonti, la nascita dei Gruppi giovanili della Democrazia cristiana, dal 1953 rinominati Movimento giovanile Dc, dalle origini nel 1944 al 1955, segnato dalla segreteria di Amintore Fanfani. L'intento č precisare alcune informazioni non corrette riportate da ricerche precedenti e dalla memorialistica, analizzando il dibattito interno all'organizzazione in alcuni tornanti decisivi del primo decennio repubblicano, mettendo in luce elementi di originalitŕ rispetto ai temi prevalenti fra gli "adulti" e offrendo, per quanto possibile, uno sguardo a livello europeo dell'associazionismo giovanile democristiano.

This essay explores the food discourse elaborated by Chinese intellectuals in the Song period (960–1279) and puts forward a hypothesis concerning its social function. Starting from the Bourdieusian...

We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $\lambda_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $\lambda\ge C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1\ll \lambda\ll n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $\lambda$ larger than a constant) critical points are either very close to the unknown vector ${\boldsymbol u}$, or are confined in a band of width $\Theta(\lambda^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${\boldsymbol u}$. For local maxima, this band shrinks to be of size $\Theta(\lambda^{-1/(k-2)})$. These `uninformative' local maxima are likely to cause the failure of optimization algorithms., Comment: 40 pages, 20 pdf figures

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...

We consider the problem of learning an unknown function $f_{\star}$ on the $d$-dimensional sphere with respect to the square loss, given i.i.d. samples $\{(y_i,{\boldsymbol x}_i)\}_{i\le n}$ where ${\boldsymbol x}_i$ is a feature vector uniformly distributed on the sphere and $y_i=f_{\star}({\boldsymbol x}_i)+\varepsilon_i$. We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: the random features model of Rahimi-Recht (RF); the neural tangent kernel model of Jacot-Gabriel-Hongler (NT). Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and enjoy universal approximation properties when the number of neurons $N$ diverges, for a fixed dimension $d$. We consider two specific regimes: the approximation-limited regime, in which $n=\infty$ while $d$ and $N$ are large but finite; and the sample size-limited regime in which $N=\infty$ while $d$ and $n$ are large but finite. In the first regime we prove that if $d^{\ell + \delta} \le N\le d^{\ell+1-\delta}$ for small $\delta > 0$, then \RF\, effectively fits a degree-$\ell$ polynomial in the raw features, and \NT\, fits a degree-$(\ell+1)$ polynomial. In the second regime, both RF and NT reduce to kernel methods with rotationally invariant kernels. We prove that, if the number of samples is $d^{\ell + \delta} \le n \le d^{\ell +1-\delta}$, then kernel methods can fit at most a a degree-$\ell$ polynomial in the raw features. This lower bound is achieved by kernel ridge regression. Optimal prediction error is achieved for vanishing ridge regularization., Comment: 65 pages; 17 pdf figures

Given a probability measure $\mu$ over ${\mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stochastic localization argument to prove a general decomposition result of this type. In Eldan's theorem, the `number of components' is characterized by the entropy of the mixture, and `closeness to product' is characterized by the covariance matrix of each component. We present an elementary proof of Eldan's theorem which makes use of an information theory (or estimation theory) interpretation. The proof is analogous to the one of an earlier decomposition result known as the `pinning lemma.', Comment: 8 pages; v2 corrects an annoying typo in the statement of the main theorem

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Zhang Tongzhi (1875–1948) was a Chinese scholar, educator, and gourmet from Nanjing, today the capital of Jiangsu province. In 1947, he published the List of Jinling's Delicacies, in which he described and extolled the best raw ingredients, processed foods, and cooked dishes he had tasted in Nanjing throughout his life. In this article, I offer a translation of twenty-four of the sixty-one entries of the List and I read them as portraying not only the city's delicacies but also the practice of gastronomy, of Zhang as well as of the people of Nanjing. As a social and consumption practice, gastronomy can be defined, with Priscilla Parkhurst Ferguson, as “the systematic, socially valorized pursuit of culinary excellence.” Yet gastronomy, in the context of Western and especially French culinary history, has mainly been analyzed as a textualized discourse. While I do not challenge the idea that gastronomic words matter, in this article I suggest that gastronomic practice matters, too, and that, as far as the development of Chinese culinary culture is concerned, it may have mattered more than words. The List affords us glimpses into a little-known foodscape and it vividly captures this “applied” dimension of gastronomy. My hope is that this text will also stimulate more comparative thinking about culinary cultures and their sociohistorical development.

In phase retrieval we want to recover an unknown signal $\boldsymbol x\in\mathbb C^d$ from $n$ quadratic measurements of the form $y_i = |\langle{\boldsymbol a}_i,{\boldsymbol x}\rangle|^2+w_i$ where $\boldsymbol a_i\in \mathbb C^d$ are known sensing vectors and $w_i$ is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements $n$ needed to produce an estimator $\hat{\boldsymbol x}(\boldsymbol y)$ that is positively correlated with the signal $\boldsymbol x$? We consider the case of Gaussian vectors $\boldsymbol a_i$. We prove that - in the high-dimensional limit - a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For $n\le d-o(d)$ no estimator can do significantly better than random and achieve a strictly positive correlation. For $n\ge d+o(d)$ a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theory arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper and lower bound generalize beyond phase retrieval to measurements $y_i$ produced according to a generalized linear model. As a byproduct of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm., Comment: 63 pages, 3 figures, presented at COLT'18 and accepted at Foundations of Computational Mathematics

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the SOS approach does not scale well to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.

This article explores attitudes toward the durian, a Southeast Asian fruit that famously arouses emotions as divergent as enticement and revulsion. Starting from the idea that tastes and feelings are historical phenomena, it argues that Western attitudes toward the durian took shape, developed, and varied under specific sociocultural circumstances. It reviews travel narratives and other relevant materials, and describes different and seemingly contradictory phases of the relationship between Westerners and the Southeast Asian fruit. In discussing the historical accounts, it suggests that such changes in attitudes are strictly related to the sociohistorical contexts in which they emerged, namely the cultural shifts involved by what Norbert Elias has described as the civilizing process (in Europe) and the different social structures brought in by colonialism (in Southeast Asia). These shifts and structures impacted on the dynamics of taste formation, and, in turn, investigating such dynamics may she...

Early identification of individuals at risk for chronic diseases is of significant clinical value. Early detection provides the opportunity to slow the pace of a condition, and thus help individuals to improve or maintain their quality of life. Additionally, it can lessen the financial burden on health insurers and self-insured employers. As a solution to mitigate the rise in chronic conditions and related costs, an increasing number of employers have recently begun using wellness programs, which typically involve an annual health risk assessment. Unfortunately, these risk assessments have low detection capability, as they should be low-cost and hence rely on collecting relatively few basic biomarkers. Thus one may ask, how can we select a low-cost set of biomarkers that would be the most predictive of multiple chronic diseases? In this paper, we propose a statistical data-driven method to address this challenge by minimizing the number of biomarkers in the screening procedure while maximizing the predictive power over a broad spectrum of diseases. Our solution uses multi-task learning and group dimensionality reduction from machine learning and statistics. We provide empirical validation of the proposed solution using data from two different electronic medical records systems, with comparisons over a statistical benchmark.

Early prediction of preeclampsia is challenging because of poorly understood causes, various risk factors, and likely multiple pathogenic phenotypes of preeclampsia. Statistical learning methods are well-equipped to deal with a large number of variables, such as patients' clinical and laboratory data, and to select the most informative features automatically.Our objective was to use statistical learning methods to analyze all available clinical and laboratory data that were obtained during routine prenatal visits in early pregnancy and to use them to develop a prediction model for preeclampsia.This was a retrospective cohort study that used data from 16,370 births at Lucile Packard Children Hospital at Stanford, CA, from April 2014 to January 2018. Two statistical learning algorithms were used to build a predictive model: (1) elastic net and (2) gradient boosting algorithm. Models for all preeclampsia and early-onset preeclampsia (34 weeks gestation) were fitted with the use of patient data that were available at16 weeks gestational age. The 67 variables that were considered in the models included maternal characteristics, medical history, routine prenatal laboratory results, and medication intake. The area under the receiver operator curve, true-positive rate, and false-positive rate were assessed via cross-validation.Using the elastic net algorithm, we developed a prediction model that contained a subset of the most informative features from all variables. The obtained prediction model for preeclampsia yielded an area under the curve of 0.79 (95% confidence interval, 0.75-0.83), sensitivity of 45.2%, and false-positive rate of 8.1%. The prediction model for early-onset preeclampsia achieved an area under the curve of 0.89 (95% confidence interval, 0.84-0.95), true-positive rate of 72.3%, and false-positive rate of 8.8%.Statistical learning methods in a retrospective cohort study automatically identified a set of significant features for prediction and yielded high prediction performance for preeclampsia risk from routine early pregnancy information.

Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors $x_i \in {\mathbb R}^p$ are obtained by applying a linear transform to a vector of i.i.d. entries, $x_i = \Sigma^{1/2} z_i$ (with $z_i \in {\mathbb R}^p$); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, $x_i = \varphi(W z_i)$ (with $z_i \in {\mathbb R}^d$, $W \in {\mathbb R}^{p \times d}$ a matrix of i.i.d. entries, and $\varphi$ an activation function acting componentwise on $W z_i$). We recover -- in a precise quantitative way -- several phenomena that have been observed in large-scale neural networks and kernel machines, including the "double descent" behavior of the prediction risk, and the potential benefits of overparametrization., Comment: 68 pages; 16 figures. This revision contains non-asymptotic version of earlier results, and results for general coefficients

We consider the problem of estimating a vector of discrete variables $(\theta_1,\cdots,\theta_n)$, based on noisy observations $Y_{uv}$ of the pairs $(\theta_u,\theta_v)$ on the edges of a graph $G=([n],E)$. This setting comprises a broad family of statistical estimation problems, including group synchronization on graphs, community detection, and low-rank matrix estimation. A large body of theoretical work has established sharp thresholds for weak and exact recovery, and sharp characterizations of the optimal reconstruction accuracy in such models, focusing however on the special case of Erd\"os--R\'enyi-type random graphs. The single most important finding of this line of work is the ubiquity of an information-computation gap. Namely, for many models of interest, a large gap is found between the optimal accuracy achievable by any statistical method, and the optimal accuracy achieved by known polynomial-time algorithms. Moreover, this gap is generally believed to be robust to small amounts of additional side information revealed about the $\theta_i$'s. How does the structure of the graph $G$ affect this picture? Is the information-computation gap a general phenomenon or does it only apply to specific families of graphs? We prove that the picture is dramatically different for graph sequences converging to amenable graphs (including, for instance, $d$-dimensional grids). We consider a model in which an arbitrarily small fraction of the vertex labels is revealed, and show that a linear-time local algorithm can achieve reconstruction accuracy that is arbitrarily close to the information-theoretic optimum. We contrast this to the case of random graphs. Indeed, focusing on group synchronization on random regular graphs, we prove that the information-computation gap still persists even when a small amount of side information is revealed., Comment: Stronger results, improved presentation. The transitivity assumption on the limiting graph is removed. Instead, we introduce and use the notion of a `tame' random rooted graph. 40 pages

Most high-dimensional estimation methods propose to minimize a cost function (empirical risk) that is a sum of losses associated to each data point (each example). In this paper, we focus on the case of nonconvex losses. Classical empirical process theory implies uniform convergence of the empirical (or sample) risk to the population risk. While under additional assumptions, uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. ¶ In order to capture the complexity of computing M-estimators, we study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we are able to establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as nonconvex binary classification, robust regression and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. ¶ We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provides a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the parameters vector (modulo logarithmic factors), then a suitable uniform convergence result holds. We apply this result to nonconvex binary classification and robust regression in very high-dimension.

Multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field. Given a set of null hypotheses $\mathcal{H}(n) = (H_1,\dotsc, H_n)$, Benjamini and Hochberg introduced the false discovery rate (FDR), which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. Nowadays FDR is the criterion of choice for large scale multiple hypothesis testing. In this paper we consider the problem of controlling FDR in an "online manner". Concretely, we consider an ordered --possibly infinite-- sequence of null hypotheses $\mathcal{H} = (H_1,H_2,H_3,\dots )$ where, at each step $i$, the statistician must decide whether to reject hypothesis $H_i$ having access only to the previous decisions. This model was introduced by Foster and Stine. We study a class of "generalized alpha-investing" procedures and prove that any rule in this class controls online FDR, provided $p$-values corresponding to true nulls are independent from the other $p$-values. (Earlier work only established mFDR control.) Next, we obtain conditions under which generalized alpha-investing controls FDR in the presence of general $p$-values dependencies. Finally, we develop a modified set of procedures that also allow to control the false discovery exceedance (the tail of the proportion of false discoveries). Numerical simulations and analytical results indicate that online procedures do not incur a large loss in statistical power with respect to offline approaches, such as Benjamini-Hochberg., Comment: 44 pages, 9 figures, to appear in Annals of Statistics

Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this paper we consider a simple case, namely two-layers neural networks, and prove that -in a suitable scaling limit- SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearly ideal generalization error. This description allows to 'average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD., Comment: 103 pages

Let ${\boldsymbol A}\in{\mathbb R}^{n\times n}$ be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing $\langle{\boldsymbol \sigma},{\boldsymbol A}{\boldsymbol \sigma}\rangle$ over binary vectors ${\boldsymbol \sigma}\in\{+1,-1\}^n$. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any $\varepsilon>0$, outputs ${\boldsymbol \sigma}_*\in\{-1,+1\}^n$ such that $\langle{\boldsymbol \sigma}_*,{\boldsymbol A}{\boldsymbol \sigma}_*\rangle$ is at least $(1-\varepsilon)$ of the optimum value, with probability converging to one as $n\to\infty$. The algorithm's time complexity is $C(\varepsilon)\, n^2$. It is a message-passing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations., Comment: 27 pages

We consider the Sherrington–Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as “${\mathbb{Z}}_{2}$ synchronization.” Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless–Anderson–Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e., the noise level is small enough in synchronization). Namely, we prove that the scaled $\ell _{2}$ distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. Our proof technique is based on upper bounding the expected number of critical points of the TAP free energy using the Kac–Rice formula.

We give explicit parametrizations of the algebraic tori $$\mathbb {T}_{n}$$Tn over any finite field $$\mathbb {F}_{q}$$Fq for any prime power $$n$$n. Applying the construction for $$n=3$$n=3 to a quadratic field $$\mathbb {F}_{q^2}$$Fq2 we show that the set of $$\mathbb {F}_q$$Fq-rational points of the torus $$\mathbb {T}_{6}$$T6 is birationally equivalent to the affine part of a Singer arc in $$\mathbb {P}^2(\mathbb {F}_{q^2})$$P2(Fq2). This gives a simple, yet efficient compression and decompression algorithm from $$\mathbb {T}_{6}(\mathbb {F}_{q})$$T6(Fq) to $$\mathbb {A}^2(\mathbb {F}_{q})$$A2(Fq) that can be substituted in the faster implementation of CEILIDH (Granger et al., in Algorithmic number theory, pp 235---249, Springer, Berlin, 2004) achieving a theoretical 30 % speedup and that is also cheaper than the recently proposed factor-$$6$$6 compression technique in Karabina (IEEE Trans Inf Theory 58(5):3293---3304, 2012). The compression methods here presented have a wide class of applications to public-key and pairing-based cryptography over any finite field.

This paper studies a class of probabilistic models on graphs, where edge variables depend on incident node variables through a fixed probability kernel. The class includes planted constraint satisfaction problems (CSPs), as well as more general structures motivated by coding and community clustering problems. It is shown that under mild assumptions on the kernel and for sparse random graphs, the conditional entropy of the node variables given the edge variables concentrates around a deterministic threshold. This implies in particular the concentration of the number of solutions in a broad class of planted CSPs, the existence of a threshold function for the disassortative stochastic block model, and the proof of a conjecture on parity check codes. It also establishes new connections among coding, clustering and satisfiability.

Il presente lavoro indaga il rapporto tra l'arte e il diritto civile dalla prospettiva del trasferimento di proprietà dell'opera d'arte. Quest'analisi viene condotta dall'Autore con particolare riferimento alle opere della c.d. arte contemporanea: ossia quelle forme artistiche che si diffondono durante il secolo scorso e quello attuale.Tali opere consistono, a differenza di quelle in cui prende corpo la c.d. arte classica, non solo in manufatti, come ad es. la scultura o il disegno, ma anche in opere c.d. immateriali, come ad es. la performance artistica. Questa differenza pone diverse questioni sul fronte normativo. L'Autore si sofferma sulla diversa rilevanza giuridica che assume il trasferimento di proprietà delle opere materiali e il trasferimentodi proprietà della performance artistica. Quest'analisi viene effettuata sia sul piano della fattispecie sia su quello della tutela e prende corpo, nel primo caso, nell'indagine dell'incidenza delle caratteristiche dell'opera d'arte sulle regole del contratto di vendita, e, nel secondo caso, nella lettura della performance artistica dalla prospettiva del contratto d'opera intellettuale e della cessione del credito.

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erd\H{o}s-Renyi random graphs with bounded average degree $d>1$, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor $2d^2/(2d^2+d-1)$ of the upper bound. In particular, the local algorithm is at most $8/9$ suboptimal, and $1+O(1/d)$ suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erd\H{o}s-Renyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model., Comment: 48 pages, 1 pdf figure

We consider the problem of reconstructing a vector x 0 ∊ Rn from noisy linear observations y = Ax o + w, where A ∊ Rm×n is a known operator and w is a noise vector, using the elastic net method. Assuming that A is random with independent and identically distributed entries, and under suitable moment conditions, we prove the following universality result. In the high-dimensional asymptotics n→∞ and m/n → δ > 0, the normalized error of the elastic net minimizer converges in probability to a limit, that does not depend on the exact distribution that the entries are drawn from. We also provide an explicit formula for the limit.

Consider an Erdos---Renyi random graph in which each edge is present independently with probability $$1/2$$1/2, except for a subset $$\mathsf{C}_N$$CN of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of Dekel et al. (ANALCO. SIAM, pp 67---75, 2011) provably identifies the clique $$\mathsf{C}_N$$CN in linear time, provided $$|\mathsf{C}_N|\ge 1.261\sqrt{N}$$|CN|?1.261N. Spectral methods can be shown to fail on cliques smaller than $$\sqrt{N}$$N. In this paper we describe a nearly linear-time algorithm that succeeds with high probability for $$|\mathsf{C}_N|\ge (1+{\varepsilon })\sqrt{N/e}$$|CN|?(1+?)N/e for any $${\varepsilon }>0$$?>0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on $$N$$N vertices). For large-girth regular graphs of degree $$(\varDelta +1)$$(Δ+1) we prove that so-called local algorithms succeed if $$|\mathsf{C}_N|\ge (1+{\varepsilon })N/\sqrt{e\varDelta }$$|CN|?(1+?)N/eΔ and fail if $$|\mathsf{C}_N|\le (1-{\varepsilon })N/\sqrt{e\varDelta }$$|CN|≤(1-?)N/eΔ.

We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. \cite{KrzakalaEtAl}, message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate $\delta$ exceeds the (upper) R\'enyi information dimension of the signal, $\uRenyi(p_X)$. More precisely, for a sequence of signals of diverging dimension $n$ whose empirical distribution converges to $p_X$, reconstruction is with high probability successful from $\uRenyi(p_X)\, n+o(n)$ measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension $n$ and $k(n)$ non-zero entries, this implies reconstruction from $k(n)+o(n)$ measurements. For `discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from $o(n)$ measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal $p_X$., Comment: 60 pages, 7 figures, Sections 3,5 and Appendices A,B are added. The stability constant is quantified (cf Theorem 1.7)

We consider a class of approximated message passing (AMP) algorithms and characterize their high-dimensional behavior in terms of a suitable state evolution recursion. Our proof applies to Gaussian matrices with independent but not necessarily identically distributed entries. It covers --in particular-- the analysis of generalized AMP, introduced by Rangan, and of AMP reconstruction in compressed sensing with spatially coupled sensing matrices. The proof technique builds on the one of [BM11], while simplifying and generalizing several steps., Comment: 29 pages, 1 figure, minor updates in citations

The binary deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Input bits are deleted independently with probability d, and when they are not deleted, they are not affected by the channel. Despite significant effort, little is known about the capacity of this channel and even less about optimal coding schemes. In this paper, we develop a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute three leading terms of this expansion, and find an input distribution that achieves capacity up to this order. This constitutes the first optimal random coding result for the deletion channel. The key idea employed is the following: We understand perfectly the deletion channel with deletion probability d=0. It has capacity 1 and the optimal input distribution is iid Bernoulli (1/2). It is natural to expect that the channel with small deletion probabilities has a capacity that varies smoothly with d, and that the optimal input distribution is obtained by smoothly perturbing the iid Bernoulli (1/2) process. Our results show that this is indeed the case.

Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to approximate message passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex penalization-the firm shrinkage nonlinearity and the minimax nonlinearity-and also nonscalar denoisers-block thresholding, monotone regression, and total variation minimization. Let the variables e = k/N and δ = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x0 according to y=Ax0. Here, A is an n×N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve δ = δ(e) separating successful from unsuccessful reconstruction of x0 by AMP is given by δ = M(e|Denoiser) where M(e|Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem. We prove that this formula follows from state evolution and present numerical results validating it in a wide range of settings. The above formula generates numerous new insights, both in the scalar and in the nonscalar cases.

Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $\alpha$. Assume that the probability that $F$ is unsatisfiable is $O(1/\log(n)^{1+\e})$ for $\e>0$. We show that (possibly excluding a countable set of `exceptional' $\alpha$'s) the number of solutions concentrate in the logarithmic scale, i.e., there exists a non-random function $\phi(\alpha)$ such that, for any $\delta>0$, $(1/n)\log Z(F)\in [\phi-\delta,\phi+\delta]$ with high probability. In particular, the assumption holds for all $\alpha

We consider communication over a noisy network under randomized linear network coding. Possible error mechanisms include node- or link-failures, Byzantine behavior of nodes, or an overestimate of the network min-cut. Building on the work of Kotter and Kschischang, we introduce a systematic oblivious random channel model. Within this model, codewords contain a header (this is the systematic part). The header effectively records the coefficients of the linear encoding functions, thus simplifying the decoding task. Under this constraint, errors are modeled as random low-rank perturbations of the transmitted codeword. We compute the capacity of this channel and we define an error-correction scheme based on random sparse graphs and a low-complexity decoding algorithm. By optimizing over the code degree profile, we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability.

We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing.

Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family, and are surprisingly well-suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that, when the `statistical noise' is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several `detection thresholds,' as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins, and use non-rigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems., Comment: 71 pages, 24 pdf figures

We consider the problem of learning a coefficient vector x_0\in R^N from noisy linear observation y=Ax_0+w \in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x'. In this case, a popular approach consists in solving an L1-penalized least squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN). For sequences of matrices A of increasing dimensions, with independent gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications., 43 pages, 5 figures (v3 rectifies some inconsistencies in the formulation of auxiliary lemmas)

We use a generalization of the Lindeberg principle developed by S. Chatterjee to prove universality properties for various problems in communications, statistical learning and random matrix theory. We also show that these systems can be viewed as the limiting case of a properly defined sparse system. The latter result is useful when the sparse systems are easier to analyze than their dense counterparts. The list of problems we consider is by no means exhaustive. We believe that the ideas can be used in many other problems relevant for information theory.

Approximate message passing algorithms proved to be extremely effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper we provide the first rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with large number of short loops in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory., Comment: 41 pages

We consider the Ising model with inverse temperature β and without external field on sequences of graphs Gn which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weakly converges to the symmetric mixture of the Ising model with + boundary conditions and the − boundary conditions on the k-regular tree with inverse temperature β. In the case where the graphs Gn are expanders we derive a more detailed understanding by showing convergence of the Ising measure conditional on positive magnetization (sum of spins) to the + measure on the tree.

Let M be a random (alpha n) x n matrix of rank r<, Comment: 30 pages, 1 figure, journal version (v1, v2: Conference version ISIT 2009)

There is a fundamental relationship between belief propagation and maximum a posteriori decoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper. This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper-bound on the maximum a posteriori threshold for sparse graph codes. In situations where this bound is tight, the extrinsic soft bit estimates delivered by the belief propagation decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold. More generally, it is conjectured that the fundamental relationship between the maximum a posteriori and the belief propagation decoder which was observed for transmission over the binary erasure channel carries over to the general case. We finally demonstrate that in order for the design rate of an ensemble to approach the capacity under belief propagation decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel., 27 pages, 46 ps figures

We study a simple game-theoretic model for the spread of an innovation in a network. The diffiusion of the innovation is modeled as the dynamics of a coordination game in which the adoption of a common strategy between players has a higher payoff. Classical results in game theory provide a simple condition for the innovation to spread through the network. The present paper characterizes the rate of convergence as a function of graph structure. In particular, we derive a dichotomy between well-connected (e.g. random) graphs that show slow convergence and poorly connected, low dimensional graphs that show fast convergence.

There is a fundamental relationship between belief propagation and maximum a posteriori decoding. A decoding algorithm, which is called the Maxwell decoder, is introduced and provides a constructive description of this relationship. Both the algorithm itself and the analysis of the new decoder are reminiscent of the Maxwell construction in thermodynamics. This paper investigates in detail the case of transmission over the binary erasure channel, while the extension to general binary memoryless channels is discussed in a companion paper.

One of the largest sectors of the European fixed-income market consists of covered bonds, a financing mechanism for bank originators of residential mortgages and other assets. The term “covered bonds” comes from the “cover pool” used as collateral. The loans remain on the bank’s balance sheet. In this article, the authors define covered bonds and briefly describe their markets in Europe and the United States. They go on to explain the significance of the July 2008 actions to jump-start the covered bond market in the United States by the FDIC, Treasury, and U.S. Rep. Scott Garrett (R-NJ). While it is unlikely that covered bonds will be a major source of funding for U.S. residential mortgages, they will undoubtedly find some place among alternative funding sources.

We explain how to optimize finite-length LDPC codes for transmission over the binary erasure channel. Our approach relies on an analytic approximation of the erasure probability. This is in turn based on a finite-length scaling result to model large scale erasures and a union bound involving minimal stopping sets to take into account small error events. We show that the performances of optimized ensembles as observed in simulations are well described by our approximation. Although we only address the case of transmission over the binary erasure channel, our method should be applicable to a more general setting.