arXiv subjects

Jonas Kahn

Publications and source records attributed to Jonas Kahn.

At least 19 recordsLinked to original sources

arXiv2024

Geodesics in planar Poisson roads random metric

We study the structure of geodesics in the fractal random metric constructed by Kendall from a self-similar Poisson process of roads (i.e, lines with speed limits) in $\mathbb{R}^2$. In particular, we prove a conjecture of Kendall stating that geodesics do not pause en route, i.e, use roads of arbitrary small speed except at their endpoints. It follows that the geodesic frame of $\left(\mathbb{R}^2,T\right)$ is the set of points on roads. We also consider geodesic stars and hubs, and give a complete description of the local structure of geodesics around points on roads. Notably, we prove that leaving a road by driving off-road is never geodesic.

math.PR↗

arXiv2021

Projected Least-Squares Quantum Process Tomography

We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propose a novel, two-step method for projecting these estimators onto the set of matrices representing physical quantum channels, and a fast numerical implementation in the form of the hyperplane intersection projection algorithm. We provide rigorous, non-asymptotic concentration bounds, sampling complexities and confidence regions for the Frobenius and trace-norm error of the estimators. For the Frobenius error, the bounds are linear in the rank of the Choi matrix, and for low ranks, they improve the error rates of the least squares estimator by a factor $d^2$, where $d$ is the system dimension. We illustrate the method with numerical experiments involving channels on systems with up to 7 qubits, and find that PLS has highly competitive accuracy and computational tractability.

quant-ph↗

arXiv2020

Sampling Rates for $\ell^1$-Synthesis

This work investigates the problem of signal recovery from undersampled noisy sub-Gaussian measurements under the assumption of a synthesis-based sparsity model. Solving the $\ell^1$-synthesis basis pursuit allows for a simultaneous estimation of a coefficient representation as well as the sought-for signal. However, due to linear dependencies within redundant dictionary atoms it might be impossible to identify a specific representation vector, although the actual signal is still successfully recovered. The present manuscript studies both estimation problems from a non-uniform, signal-dependent perspective. By utilizing recent results on the convex geometry of linear inverse problems, the sampling rates describing the phase transitions of each formulation are identified. In both cases, they are given by the conic Gaussian mean width of an $\ell^1$-descent cone that is linearly transformed by the dictionary. In general, this expression does not allow a simple calculation by following the polarity-based approach commonly found in the literature. Hence, two upper bounds involving the sparsity of coefficient representations are provided: The first one is based on a local condition number and the second one on a geometric analysis that makes use of the thinness of high-dimensional polyhedral cones with not too many generators. It is furthermore revealed that both recovery problems can differ dramatically with respect to robustness to measurement noise -- a fact that seems to have gone unnoticed in most of the related literature. All insights are carefully undermined by numerical simulations.

cs.IT↗

arXiv2019

Approximation of curves with piecewise constant or piecewise linear functions

In this paper we compute the Hausdorff distance between sets of continuous curves and sets of piecewise constant or linear discretizations. These sets are Sobolev balls given by the continuous or discrete $L^p$-norm of the derivatives. We detail the suitable discretization or smoothing procedure which are preservative in the sense of these norms. Finally we exhibit the link between Eulerian numbers and the uniformly space knots B-spline used for smoothing.

math.NA↗

arXiv2018

Parameter recovery in two-component contamination mixtures: the $\mathbb{L}^2$ strategy

In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^\star =(1-λ^\star) ϕ+ λ^\star ϕ(.-μ^\star)$, where the shape $ϕ$ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(λ^\star,μ^\star)$. In particular, we prove that the classical parametric rate $1/\sqrt{n}$ cannot be reached when at least one of these parameters is allowed to tend to $0$ with $n$.

math.ST↗

arXiv2018

Fast state tomography with optimal error bounds

Projected least squares (PLS) is an intuitive and numerically cheap technique for quantum state tomography. The method first computes the least-squares estimator (or a linear inversion estimator) and then projects the initial estimate onto the space of states. The main result of this paper equips this point estimator with a rigorous, non-asymptotic confidence region expressed in terms of the trace distance. The analysis holds for a variety of measurements, including 2-designs and Pauli measurements. The sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and -- in the case of measuring the uniform POVM -- saturates fundamental lower bounds.The results are derived by reinterpreting the least-squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality. The theory is supported by numerical simulations for mutually unbiased bases, Pauli observables, and Pauli basis measurements.

quant-ph↗

arXiv2018

3/4-discrete optimal transport

This paper deals with the 3/4-discrete 2-Wasserstein optimal transport between two measures, where one is supported by a set of segment and the other one is supported by a set of Dirac masses. We select the most suitable optimization procedure that computes the optimal transport and provide numerical examples of approximation of cloud data by segments.

math.NA↗

arXiv2018

Optimal Transport Approximation of 2-Dimensional Measures

We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves for instance. The proposed principle and algorithm are a natural generalization of previous results revolving around the generation of blue-noise point distributions, such as Lloyd's algorithm or more advanced techniques based on power diagrams. We analyze the convergence properties and propose new approaches to accelerate the generation of point distributions. We also design new algorithms to project curves onto spaces of curves with bounded length and curvature or speed and acceleration. We illustrate the algorithm's interest through applications in advanced sampling theory, non-photorealistic rendering and path planning.

math.NA↗

arXiv2018

Recursive nearest agglomeration (ReNA): fast clustering for approximation of structured signals

In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure, such as with images. We focus on this setting and investigate feature clustering schemes for data reductions that capture this structure. An impediment to fast dimension reduction is that good clustering comes with large algorithmic costs. We address it by contributing a linear-time agglomerative clustering scheme, Recursive Nearest Agglomeration (ReNA). Unlike existing fast agglomerative schemes, it avoids the creation of giant clusters. We empirically validate that it approximates the data as well as traditional variance-minimizing clustering schemes that have a quadratic complexity. In addition, we analyze signal approximation with feature clustering and show that it can remove noise, improving subsequent analysis steps. As a consequence, data reduction by clustering features with ReNA yields very fast and accurate models, enabling to process large datasets on budget. Our theoretical analysis is backed by extensive experiments on publicly-available data that illustrate the computation efficiency and the denoising properties of the resulting dimension reduction scheme.

stat.ML↗

arXiv2018

Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure

This paper aims at determining under which conditions the semi-discrete optimal transport is twice differentiable with respect to the parameters of the discrete measure and exhibits numerical applications. The discussion focuses on minimal conditions on the background measure to ensure differentiability. We provide numerical illustrations in stippling and blue noise problems.

math.NA↗

arXiv2017

Multiview Attenuation Estimation and Correction

Measuring attenuation coefficients is a fundamental problem that can be solved with diverse techniques such as X-ray or optical tomography and lidar. We propose a novel approach based on the observation of a sample from a few different angles. This principle can be used in existing devices such as lidar or various types of fluorescence microscopes. It is based on the resolution of a nonlinear inverse problem. We propose a specific computational approach to solve it and show the well-foundedness of the approach on simulated data. Some of the tools developed are of independent interest. In particular we propose an efficient method to correct attenuation defects, new robust solvers for the lidar equation as well as new efficient algorithms to compute the Lambert W function and the proximal operator of the logsumexp function in dimension 2.

math.OC↗

arXiv2016

Improper poisson line process as sirsn in any dimension

Aldous has introduced a notion of scale-invariant random spatial network (SIRSN) as a mathematical formalization of road networks. Intuitively, those are random processes that assign a route between each pair of points in Euclidean space, while being invariant under rotation, translation, and change of scale, and such that the routes are not too long and mainly on "main roads". The only known example was somewhat artificial since invariance had to be added at the end of the construction. We prove that the network of geodesics in the random metric space generated by a Poisson line process marked by speeds according to a power law is a SIRSN, in any dimension. Along the way, we establish bounds comparing Euclidean balls and balls for the random metric space. We also prove that in dimension more than two, the geodesics have "many directions" near each point where they are not straight.

math.PR↗

arXiv2015

Fast clustering for scalable statistical analysis on structured images

The use of brain images as markers for diseases or behavioral differences is challenged by the small effects size and the ensuing lack of power, an issue that has incited researchers to rely more systematically on large cohorts. Coupled with resolution increases, this leads to very large datasets. A striking example in the case of brain imaging is that of the Human Connectome Project: 20 Terabytes of data and growing. The resulting data deluge poses severe challenges regarding the tractability of some processing steps (discriminant analysis, multivariate models) due to the memory demands posed by these data. In this work, we revisit dimension reduction approaches, such as random projections, with the aim of replacing costly function evaluations by cheaper ones while decreasing the memory requirements. Specifically, we investigate the use of alternate schemes, based on fast clustering, that are well suited for signals exhibiting a strong spatial structure, such as anatomical and functional brain images. Our contribution is twofold: i) we propose a linear-time clustering scheme that bypasses the percolation issues inherent in these algorithms and thus provides compressions nearly as good as traditional quadratic-complexity variance-minimizing clustering schemes, ii) we show that cluster-based compression can have the virtuous effect of removing high-frequency noise, actually improving subsequent estimations steps. As a consequence, the proposed approach yields very accurate models on several large-scale problems yet with impressive gains in computational efficiency, making it possible to analyze large datasets.

stat.ML↗

arXiv2015

A projection algorithm on measures sets

We consider the problem of projecting a probability measure $\pi$ on a set $\mathcal{M}\_N$ of Radon measures. The projection is defined as a solution of the following variational problem:\begin{equation*}\inf\_{\mu\in \mathcal{M}\_N} \|h\star (\mu - \pi)\|\_2^2,\end{equation*}where $h\in L^2(\Omega)$ is a kernel, $\Omega\subset \R^d$ and $\star$ denotes the convolution operator.To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with $N$ dots) or continuous line drawing (representing an image with a continuous line).We provide a necessary and sufficient condition on the sequence $(\mathcal{M}\_N)\_{N\in \N}$ that ensures weak convergence of the projections $(\mu^*\_N)\_{N\in \N}$ to $\pi$.We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings.

math.NA↗

arXiv2015

Optimal rates for finite mixture estimation

We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with $m_0$ components, the optimal local minimax rate of estimation of a mixing distribution with $m$ components is $n^{-1/(4(m-m_0) + 2)}$. This corrects a previous paper by Chen (1995) in The Annals of Statistics. By contrast, it turns out that there are estimators with a (non-uniform) pointwise rate of estimation of $n^{-1/2}$ for all mixing distributions with a finite number of components.

math.ST↗

arXiv2015

Minimax rates for finite mixture estimation

We prove that under some regularity and strong identifiability conditions, around a mixing distribution with $m_0$ components, the optimal local minimax rate of estimation of a mixture with $m$ components is $n^{-1/(4(m-m_0) + 2)}$. This corrects a previous paper by Chen (1995) in The Annals of Statistics.

math.ST↗

arXiv2014

Gradient waveform design for variable density sampling in Magnetic Resonance Imaging

Fast coverage of k-space is a major concern to speed up data acquisition in Magnetic Resonance Imaging (MRI) and limit image distortions due to long echo train durations. The hardware gradient constraints (magnitude, slew rate) must be taken into account to collect a sufficient amount of samples in a minimal amount of time. However, sampling strategies (e.g., Compressed Sensing) and optimal gradient waveform design have been developed separately so far. The major flaw of existing methods is that they do not take the sampling density into account, the latter being central in sampling theory. In particular, methods using optimal control tend to agglutinate samples in high curvature areas. In this paper, we develop an iterative algorithm to project any parameterization of k-space trajectories onto the set of feasible curves that fulfills the gradient constraints. We show that our projection algorithm provides a more efficient alternative than existinf approaches and that it can be a way of reducing acquisition time while maintaining sampling density for piece-wise linear trajectories.

math.OC↗

arXiv2014

Variable density sampling with continuous trajectories. Application to MRI

Reducing acquisition time is a crucial challenge for many imaging techniques. Compressed Sensing (CS) theory offers an appealing framework to address this issue since it provides theoretical guarantees on the reconstruction of sparse signals by projection on a low dimensional linear subspace. In this paper, we focus on a setting where the imaging device allows to sense a fixed set of measurements. We first discuss the choice of an optimal sampling subspace (smallest subset) allowing perfect reconstruction of sparse signals. Its standard design relies on the random drawing of independent measurements. We discuss how to select the drawing distribution and show that a mixed strategy involving partial deterministic sampling and independent drawings can help breaking the so-called "coherence barrier". Unfortunately, independent random sampling is irrelevant for many acquisition devices owing to acquisition constraints. To overcome this limitation, the notion of Variable Density Samplers (VDS) is introduced and defined as a stochastic process with a prescribed limit empirical measure. It encompasses samplers based on independent measurements or continuous curves. The latter are crucial to extend CS results to actual applications. Our main contribution lies in two original continuous VDS. The first one relies on random walks over the acquisition space whereas the second one is heuristically driven and rests on the approximate solution of a Traveling Salesman Problem. Theoretical analysis and retrospective CS simulations in magnetic resonance imaging highlight that the TSP-based solution provides improved reconstructed images in terms of signal-to-noise ratio compared to standard sampling schemes (spiral, radial, 3D iid...).

stat.AP↗