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The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical…
We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an…
We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving…
How can we process a piece of recorded music to detect and visualize the onset of each instrument? A simple, interpretable approach is based on partially fixed nonnegative matrix factorization (NMF). Yet despite the method's simplicity,…
We introduce a simple, rank-based algorithm for inverting the sweep map on (2n,n)-Dyck paths.
Most optimization problems are notoriously hard. Considerable efforts must be spent in obtaining an optimal solution to certain instances that we encounter in the real world scenarios. Often it turns out that input instances get modified…
The 'signature method' refers to a collection of feature extraction techniques for multivariate time series, derived from the theory of controlled differential equations. There is a great deal of flexibility as to how this method can be…
Even if path planning can be solved using standard techniques from dynamic programming and control, the problem can also be approached using probabilistic inference. The algorithms that emerge using the latter framework bear some appealing…
We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their covariance functions. Similar estimates have…
In an article published in 1987 in Combinatorica \cite{MR918397}, Frieze and Jackson established a lower bound on the length of the longest induced path (and cycle) in a sparse random graph. Their bound is obtained through a rough analysis…
The auto-encoder method is a type of dimensionality reduction method. A mapping from a vector to a descriptor that represents essential information can be automatically generated from a set of vectors without any supervising information.…
Sign Gradient Descent (SignGD) is a simple yet robust optimization method, widely used in machine learning for its resilience to gradient noise and compatibility with low-precision computations. While its empirical performance is well…
We present a technique to infer lower bounds on the worst-case runtime complexity of integer programs, where in contrast to earlier work, our approach is not restricted to tail-recursion. Our technique constructs symbolic representations of…
Hyperspectral analysis has gained popularity over recent years as a way to infer what materials are displayed on a picture whose pixels consist of a mixture of spectral signatures. Computing both signatures and mixture coefficients is known…
In any graph, the maximum size of an induced path is bounded by the maximum size of a path. However, in the general case, one cannot find a converse bound, even up to an arbitrary function, as evidenced by the case of cliques. Galvin, Rival…
We develop a novel, fundamental and surprisingly simple randomized iterative method for solving consistent linear systems. Our method has six different but equivalent interpretations: sketch-and-project, constrain-and-approximate, random…
The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to…
We propose new approach to numerical study of quantum spin systems. Our method is based on a fact that one can use any set of states for the path integral as long as it is complete. We apply our method to one-dimensional quantum spin system…
We present a quantum algorithm for sampling an edge on a path between two nodes s and t in an undirected graph given as an adjacency matrix, and show that this can be done in query complexity that is asymptotically the same, up to log…
In this paper, we propose a novel approach that enhances recurrent neural networks (RNNs) by incorporating path signatures into their gating mechanisms. Our method modifies both Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU)…