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We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…
We derive new and improved non-asymptotic deviation inequalities for the sample average approximation (SAA) of an optimization problem. Our results give strong error probability bounds that are "sub-Gaussian"~even when the randomness of the…
Autoencoders are widely used for unsupervised learning and as a regularization scheme in semi-supervised learning. However, theoretical understanding of their generalization properties and of the manner in which they can assist supervised…
The learning properties of finite size polynomial Support Vector Machines are analyzed in the case of realizable classification tasks. The normalization of the high order features acts as a squeezing factor, introducing a strong anisotropy…
In this paper, we establish generalization bounds for transductive learning algorithms in the context of information theory and PAC-Bayes, covering both the random sampling and the random splitting setting. First, we show that the…
We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so.…
This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many…
The aim of this paper is to use non asymptotic bounds for the probability of rare events in the Sanov theorem, in order to study the asymptotics in conditional limit theorems (Gibbs conditioning principle for thin sets). Applications to…
For arbitrary two probability measures on real d-space with given means and variances (covariance matrices), we provide lower bounds for their total variation distance. In the one-dimensional case, a tight bound is given.
We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign…
This paper is concerned with the hard thresholding operator which sets all but the $k$ largest absolute elements of a vector to zero. We establish a {\em tight} bound to quantitatively characterize the deviation of the thresholded solution…
This article provides an original understanding of the behavior of a class of graph-oriented semi-supervised learning algorithms in the limit of large and numerous data. It is demonstrated that the intuition at the root of these methods…
Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for…
Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to lie within lens-shaped regions in the…
In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on…
We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space. We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real…
Generalization of deep neural networks remains one of the main open problems in machine learning. Previous theoretical works focused on deriving tight bounds of model complexity, while empirical works revealed that neural networks exhibit…
We provide finite sample upper and lower bounds on the Binomial tail probability which are a direct application of Sanov's theorem. We then use these to obtain high probability upper and lower bounds on the minimum of i.i.d. Binomial random…
We give a new asymptotic upper bound on the size of a code in the Grassmannian space. The bound is better than the upper bounds known previously in the entire range of distances except very large values.