Bernstein Inequality Research Articles

A Bernstein type inequality associated with wavelet bi-frame decomposition

Bernstein inequality is an essential inequality for Besov spaces. Smoothness based approaches are widely used in establishing the inequality. Yet, despite numerous studies over the last two decades, there is still little research focusing on decay-based approaches. However, motivating authors to establish inequality poses challenges, many of which can be overcome by means of the completeness of wavelet bi-frames in Lebesgue spaces and the stability of wavelet coefficients. The research has shown how wavelets with decay conditions enable descriptions of Lebesgue spaces, and in particular, the Bernstein inequality.

Weighted fractional Bernstein’s inequalities and their applications

This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on S d-1 $$\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d$$ , where Π denotes the space of all spherical polynomials of degree at most n on S d-1 and (-Δ0) r/2 is the fractional Laplacian-Beltrami operator on S d-1. A new class of doubling weights with conditions weaker than the A p condition is introduced and used to characterize completely those doubling weights w on S d-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ L p (S d-1) with 0 < p < 1 can be approximated by the de la Vallee Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights.

The strong consistency of M-estimates in linear models with extended negatively dependent errors

ABSTRACTIn this paper, we first establish the strong convergence for weighted sums of extended negatively dependent (END) random variables. Based on the strong convergence and Bernstein inequality, we obtain the strong consistency of M-estimates of the regression parameters in a linear model for END random errors under some mild moment conditions. The results generalize and improve the ones obtained in the literature to the case of END random errors.

Bernstein-type inequality for a class of dependent random matrices

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

Optimal polynomial admissible meshes on some classes of compact subsets of [formula omitted

We show that any compact subset of Rd which is the closure of a bounded star-shaped Lipschitz domain Ω, such that ∁Ω has positive reach in the sense of Federer, admits an optimal AM (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on C2 star-shaped domains.Moreover, we prove constructively the existence of an optimal AM for any K:=Ω¯⊂Rd where Ω is a bounded C1,1 domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.

Risk Minimization in the Presence of Label Noise

Matrix concentration inequalities have attracted much attention in diverse applications such as linear algebra, statistical estimation, combinatorial optimization, etc. In this paper, we present new Bernstein concentration inequalities depending only on the first moments of random matrices, whereas previous Bernstein inequalities are heavily relevant to the first and second moments. Based on those results, we analyze the empirical risk minimization in the presence of label noise. We find that many popular losses used in risk minimization can be decomposed into two parts, where the first part won't be affected and only the second part will be affected by noisy labels. We show that the influence of noisy labels on the second part can be reduced by our proposed LICS (Labeled Instance Centroid Smoothing) approach. The effectiveness of the LICS algorithm is justified both theoretically and empirically.

On the discrepancy of jittered sampling

We study the discrepancy of jittered sampling sets: such a set P⊂[0,1]d is generated for fixed m∈N by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 110dN12+12d≤EDN∗(P)≤d(logN)12N12+12d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky–Kiefer–Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points.

Some Inequalities for Trigonometric Polynomials and Fourier Coefficients

The Bernstein inequalities for trigonometric polynomials are generalized. For sums of Fourier coefficients, upper bounds with certain constants in terms of quantities characterizing structural properties of functions are obtained. Bibliography: 9 titles.

A Study of the Sequence of Norm of Derivatives (or Primitives) of Functions Depending on Their Beurling Spectrum

In this paper, we characterize the behavior of the sequence of norm of derivatives (or primitives) of functions by their Beurling spectrum in Banach spaces. The Bernstein inequality for Banach spaces is also obtained.

Dirichlet orthogonal polynomials with Laguerre weight

Let {λj}j=1∞ be a sequence of distinct positive numbers. We find explicit formulae for the orthogonal Dirichlet polynomials {ψn} formed from linear combinations of {λj−it}j=1n, associated with the Laguerre weight. Thus ∫0∞ψn(t)ψm(t)¯e−tdt=δmn. In addition, we estimate Christoffel functions and establish Markov–Bernstein inequalities.

Fast community detection by SCORE

Consider a network where the nodes split into $K$ different communities. The community labels for the nodes are unknown and it is of major interest to estimate them (i.e., community detection). Degree Corrected Block Model (DCBM) is a popular network model. How to detect communities with the DCBM is an interesting problem, where the main challenge lies in the degree heterogeneity. We propose a new approach to community detection which we call the Spectral Clustering On Ratios-of-Eigenvectors (SCORE). Compared to classical spectral methods, the main innovation is to use the entry-wise ratios between the first leading eigenvector and each of the other leading eigenvectors for clustering. Let $A$ be the adjacency matrix of the network. We first obtain the $K$ leading eigenvectors of $A$, say, $\hat{\eta}_1,\ldots,\hat{\eta}_K$, and let $\hat{R}$ be the $n\times (K-1)$ matrix such that $\hat{R}(i,k)=\hat{\eta}_{k+1}(i)/\hat{\eta}_1(i)$, $1\leq i\leq n$, $1\leq k\leq K-1$. We then use $\hat{R}$ for clustering by applying the $k$-means method. The central surprise is, the effect of degree heterogeneity is largely ancillary, and can be effectively removed by taking entry-wise ratios between $\hat{\eta}_{k+1}$ and $\hat{\eta}_1$, $1\leq k\leq K-1$. The method is successfully applied to the web blogs data and the karate club data, with error rates of $58/1222$ and $1/34$, respectively. These results are more satisfactory than those by the classical spectral methods. Additionally, compared to modularity methods, SCORE is easier to implement, computationally faster, and also has smaller error rates. We develop a theoretic framework where we show that under mild conditions, the SCORE stably yields consistent community detection. In the core of the analysis is the recent development on Random Matrix Theory (RMT), where the matrix-form Bernstein inequality is especially helpful.

Consistency of spectral clustering in stochastic block models

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

Dual lower bounds for approximate degree and Markov–Bernstein inequalities

The ε-approximate degree of a Boolean function f:{−1,1}n→{−1,1} is the minimum degree of a real polynomial that approximates f to within error ε in the ℓ∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND–OR tree for any constant ε>0. We show that this quantity is Θ(n), closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov (Theory Comput. 2013) using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek (2008). Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

On Bernstein and Erdős–Lax's inequalities for quaternionic polynomials

In this paper the well-known Bernstein's inequality for complex polynomials is extended to the quaternionic setting. We also show that the Erdős–Lax's inequality does not hold in general, but it works for a particular class of polynomials.

U-processes and preference learning.

Preference learning has caused great attention in machining learning. In this letter we propose a learning framework for pairwise loss based on empirical risk minimization of U-processes via Rademacher complexity. We first establish a uniform version of Bernstein inequality of U-processes of degree 2 via the entropy methods. Then we estimate the bound of the excess risk by using the Bernstein inequality and peeling skills. Finally, we apply the excess risk bound to the pairwise preference and derive the convergence rates of pairwise preference learning algorithms with squared loss and indicator loss by using the empirical risk minimization with respect to U-processes.

The Bernstein inequality and the Schur inequality are equivalent

It is shown that the Bernstein Inequality and the Schur Inequality are equivalent in the sense that each can be obtained from the other one with the aid of brief elementary arguments.

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

The unitary polar factor $Q=U_p$ in the polar decomposition of $Z=U_p \, H$ is the minimizer over unitary matrices $Q$ for both $\|{\rm Log}(Q^* Z)\|^2$ and its Hermitian part $\|{{\rm sym}{_{_*}}\!}({\rm Log}(Q^* Z))\|^2$ over both $\mathbb{R}$ and $\mathbb{C}$ for any given invertible matrix $Z\in\mathbb{C}^{n\times n}$ and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any $n$ and for the Frobenius matrix norm for $n\leq 3$. The result shows that the unitary polar factor is the nearest orthogonal matrix to $Z$ not only in the normwise sense but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all $z\in\mathbb{C}{\setminus}\{0\} \min_{\vartheta\in(-\pi,\pi]}{|{\rm Log}_{\mathbb{C}}(e^{-i\vartheta} z)|}^2={|{\rm log}|...

Discrete Bernstein inequalities for polynomials

Discrete Bernstein inequalities for polynomials

Noncommutative martingale concentration inequalities

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a martingale difference.

Entire [formula omitted]-functions of exponential type

We characterize the entire functions of exponential type, whose restriction to the real line is in Lp, in different ways: by the usual classical Paley–Wiener growth estimates in the complex plane, by Bernstein inequalities using derivatives or differences, by Lp-growth properties of iterated derivatives or differences, and by support properties of the Fourier image. We also establish Paley–Wiener theorems for the Fourier series of a function on the circle.