Related papers: On multi-dimensional sampling and interpolation
We provide an interpretation of entanglement based on classical correlations between measurement outcomes of complementary properties: states that have correlations beyond a certain threshold are entangled. The reverse is not true, however.…
The existence of fundamental cardinal exponential B-splines of positive real order $\sigma$ is established subject to two conditions on $\sigma$ and their construction is implemented. A sampling result for these fundamental cardinal…
Recent research has generated hope that inference scaling, such as resampling solutions until they pass verifiers like unit tests, could allow weaker models to match stronger ones. Beyond inference, this approach also enables training…
We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal…
Let I_1,...,I_n be independent but not necessarily identically distributed Bernoulli random variables, and let X_n=\sum_{j=1}^nI_j. For \nu in a bounded region, a local central limit theorem expansion of P(X_n=EX_n+\nu) is developed to any…
Resampling is an operation costly in calculation time and accuracy. It regularizes irregular sampling, replacing N data by N periodic estimations. This stage can be suppressed, using formulas built with incoming data and completed by…
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
We have exploited a variety of techniques to study the universality and stability of the scaling properties of Harper's equation, the equation for a particle moving on a tight-binding square lattice in the presence of a gauge field, when…
Both sampling a time-varying signal, and its spectral analysis are activities subjected to theoretically compelling, such as Shannon's theorem and the objectively limiting of the frequency's resolution. Usually, the signals' spectra are…
We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension.This hinges on a fully nonlinear obstacle formulation [A. M. Oberman, "The convex envelope is the solution of a…
Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result…
Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional…
We provide a description of the interpolating and sampling sequences on a space of holomorphic functions with a uniform growth restriction defined on finite Riemann surfaces.
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by…
The Shannon sampling theorem for bandlimited wide sense stationary random processes was established in 1957, which and its extensions to various random processes have been widely studied since then. However, truncation of the Shannon series…
This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert…
Complex systems are fascinating because their rich macroscopic properties emerge from the interaction of many simple parts. Understanding the building principles of these emergent phenomena in nature requires assessing natural complex…
Learning joint probability distributions on n random variables requires exponential sample size in the generic case. Here we consider the case that a temporal (or causal) order of the variables is known and that the (unknown) graph of…
Maximum Variance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the…
Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out…