This paper deals with the trace regression model where n entries or linear combinations of an unknown m1 × m2 matrix A0 corrupted by noise are observed. We propose a new nuclear-norm penalized estimator and establish general sharp oracle inequality for this arbitrary values n, m1, under condition isometry in expectation. Then method is applied to completion problem. In case, admits simple explicit form, we prove that it satisfies inequalities faster rates convergence than previous works....

We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations simple classifiers. Such could be implemented by neural networks or voting methods combining the classifiers, such as boosting and bagging. The in terms empirical distribution margin combined classifier. They based theory Gaussian processes (comparison inequalities, symmetrization method, concentration inequalities) they improve previous results Bartlett (1998) bounding $\ell_1$-norms...

Let ℱ be a class of measurable functions f:S↦[0, 1] defined on probability space (S, $\mathcal{A}$, P). Given sample (X1, …, Xn) i.i.d. random variables taking values in S with common distribution P, let Pn denote the empirical measure based Xn). We study an risk minimization problem Pnf→min , f∈ℱ. solution f̂n this problem, goal is to obtain very general upper bounds its excess $$\mathcal{E}_{P}(\hat{f}_{n}):=P\hat{f}_{n}-\inf_{f\in \mathcal{F}}Pf,$$ expressed terms relevant geometric...

Let $X,X_{1},\dots,X_{n},\dots$ be i.i.d. centered Gaussian random variables in a separable Banach space $E$ with covariance operator $\Sigma$: \[\Sigma:E^{\ast}\mapsto E,\qquad\Sigma u=\mathbb{E}\langle X,u\rangle X,\qquad u\in E^{\ast}.\] The sample $\hat{\Sigma}:E^{\ast}\mapsto E$ is defined as \[\hat{\Sigma}u:=n^{-1}\sum_{j=1}^{n}\langle X_{j},u\rangle X_{j},\qquad goal of the paper to obtain concentration inequalities and expectation bounds for norm $\Vert \hat{\Sigma}-\Sigma\Vert $...

We suggest a penalty function to be used in various problems of structural risk minimization. This is data dependent and based on the sup-norm so-called Rademacher process indexed by underlying class functions (sets). The standard complexity penalties, learning VC-dimensions classes, are conservative upper bounds (in probabilistic sense, uniformly over set all distributions) for we suggest. Thus, particular distribution training examples, one can expect better performance algorithms with...

The problem of multiple kernel learning based on penalized empirical risk minimization is discussed. complexity penalty determined jointly by the L2 norms and reproducing Hilbert space (RKHS) induced kernels with a data-driven choice regularization parameters. main focus case when total number large, but only relatively small them needed to represent target function, so that sparse. goal establish oracle inequalities for excess resulting prediction rule showing method adaptive both unknown...

Given X a random vector in R n , set 1 ..., N to be independent copies of and let Γ = √ i=1 i • e the matrix whose rows are X1 . ., XN .We obtain sharp probabilistic lower bounds on smallest singular value λ min (Γ) rather general situation, particular, under assumption that is an isotropic for which sup t∈S n-1 E| t, | 2+η ≤ L some L, η > 0. Our results imply Bai-Yin type bound holds 2, and, up log-factor, 2 as well.The hold without any additional assumptions Euclidean norm ℓ .Moreover, we...

Soit $(X, Y)$ un couple aléatoire à valeurs dans $S×T$ et de loi $P$ inconnue. Soient $(X_1, Y_1), …, (X_n, Y_n)$ des répliques i.i.d. Y)$, empirique associée $P_n$. $h_1, h_N:S↦[−1, 1]$ dictionnaire composé $N$ fonctions. Pour tout $λ∈ℝ^N$, on note $f_λ:=∑_{j=1}^Nλ_jh_j$. $ℓ:T×ℝ↦ℝ$ fonction perte donnée que l'on suppose convexe en la seconde variable. On $(ℓ•f)(x, y):=ℓ(y;f(x))$. étudie le problème minimisation du risque pénalisé suivant $$\hat{\lambda}^{\varepsilon }:=\mathop{\operatorname...

The paper develops a class of extensions the univariate quantile function to multivariate case (M-quantiles), related in certain way M-parameters probability distribution and their M-estimators. spatial (geometric) quantiles, recently introduced by Koltchinskii Dudley Chaudhuri as well regression quantiles Koenker Basset, are examples M-quantile discussed paper. We study main properties M-quantiles develop asymptotic theory empirical M-quantiles. useM-quantiles extend L-parameters...

L 2 (S, S , P) U 3 be a compact integral operator with symmetric kernel h.Let X i P N, independent S-valued random variables common probability law P. Consider the n matrix H entries À1 h(X j ), 1 < i, (this is of an empirical version replaced by measure and let denote modi®cation obtained deleting its diagonal.It proved that l distance between ordered spectrum tends to zero a.s.if only if Hilbert±Schmidt.Rates convergence distributional limit theorems for difference spectra operators (or )...

Let ℱ be a class of measurable functions on space $(S,\mathscr{S})$ with values in [0,1] and let $$P_n=n^{−1}\sum_{i=1}^nδ{X_i}$$ the empirical measure based an i.i.d. sample (X1,…,Xn) from probability distribution P $(S,\mathscr{S})$. We study behavior suprema following type: $$\sup_{r_{n}<\sigma_{P}f\leq \delta_{n}}\frac{|P_{n}f-Pf|}{\phi(\sigma_{P}f)},$$ where σPf≥Var1/2Pf ϕ is continuous, strictly increasing function ϕ(0)=0. Using Talagrand's concentration inequality for processes, we...

Let $$ Y_j=f_{\ast}(X_j)+\xi_j,\qquad j=1,\dots, n, where $X, X_1,\dots, X_n$ are i.i.d. random variables in a measurable space $(S,\mathcal{A})$ with distribution $\Pi$ and $\xi, \xi_1,\dots ,\xi_n$ ${\mathbb E}\xi=0$ independent of $(X_1,\dots, X_n).$ Given dictionary $h_1,\dots, h_{N}: S\mapsto{\mathbb R},$ let $ f_{\lambda}:=\sum_{j=1}^N \lambda_j h_j$, \lambda=(\lambda_1,\dots, \lambda_N)\in{\mathbb R}^N. $\varepsilon>0,$ define \hat\Lambda_{\varepsilon}:=\Biggl\{\lambda\in{\mathbb...

We study a problem of estimation Hermitian nonnegatively definite matrix ρ unit trace (e.g., density quantum system) based on n i.i.d. measurements (X1, Y1), …, (Xn, Yn), where Yj = tr(ρXj) + ξj, j 1, n, {Xj} being random matrices and {ξj} variables with ${\mathbb{E}}(\xi_{j}|X_{j})=0$. The estimator $$\hat{\rho}^{\varepsilon }:=\mathop{\arg\min}_{S\in{\mathcal{S}}}\Biggl[n^{-1}\sum_{j=1}^{n}\bigl(Y_{j}-\operatorname{tr}(SX_{j})\bigr)^{2}+\varepsilon \operatorname{tr}(S\log S)\Biggr] $$ is...

Soient $X,X_{1},\ldots,X_{n}$ des vecteurs gaussiens à valeurs dans un espace de Hilbert séparable $\mathbb{H}$, i.i.d. et centrés. Nous définissons l'opérateur covariance $\Sigma=\mathbb{E}(X\otimes X)$ le rang effectif $\Sigma $ \[\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma\|_{\infty}}\] où $\operatorname{tr}(\Sigma)$ est la trace of $\|\Sigma\|_{\infty }$ sa norme d'opérateur. considérons \[\hat{\Sigma}_{n}:=n^{-1}\sum_{j=1}^{n}(X_{j}\otimes X_{j})\] empirique construit...

Let $X,X_{1},\dots,X_{n}$ be i.i.d. Gaussian random variables in a separable Hilbert space $\mathbb{H}$ with zero mean and covariance operator $\Sigma=\mathbb{E}(X\otimes X)$, let $\hat{\Sigma}:=n^{-1}\sum_{j=1}^{n}(X_{j}\otimes X_{j})$ the sample (empirical) based on $(X_{1},\dots,X_{n})$. Denote by $P_{r}$ spectral projector of $\Sigma$ corresponding to its $r$th eigenvalue $\mu_{r}$ $\hat{P}_{r}$ empirical counterpart $P_{r}$. The main goal paper is obtain tight bounds...

Consider a problem of recovery smooth function (signal, image) f/spl isin//spl Fscr//spl isin/L/sub 2/([0, 1]/sup d/) passed through an unknown filter and then contaminated by noise. A typical model discussed in the paper is described stochastic differential equation dY/sub f//sup /spl epsi//(t)=(Hf)(t)dt+/spl epsi/dW(t), t/spl isin/[0, d/, epsi/>0 where H linear operator modeling W Brownian motion (sheet) The aim to recover f with asymptotically (as epsi//spl rarr/0) minimax mean integrated...

Probabilistic methods and statistical learning theory have been shown to provide approximate solutions "difficult" control problems. Unfortunately, the number of samples required in order guarantee stringent performance levels may be prohibitively large. This paper introduces bootstrap concept stopping times drastically reduce bound on achieve a level. We then apply these results obtain more efficient algorithms which probabilistically stability robustness when designing controllers for...