Albert Cohen

Albert Cohen, Mark A. Davenport, Dany Leviatan

We consider the problem of reconstructing an unknown function $f$ on a domain $X$ from samples of $f$ at $n$ randomly chosen points with respect to a given measure $ρ_X$. Given a sequence of linear spaces $(V_m)_{m>0}$ with ${\rm dim}(V_m)=m\leq n$, we study the least squares approximations from the spaces $V_m$. It is well known that such approximations can be inaccurate when $m$ is too close to $n$, even when the samples are noiseless. Our main result provides a criterion on $m$ that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in $L^2(X,ρ_X)$, is comparable to the best approximation error of $f$ by elements from $V_m$. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with $ρ_X$ being the uniform measure, and algebraic polynomials, with $ρ_X$ being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.

Markus Bachmayr, Albert Cohen, Wolfgang Dahmen

We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations separating spatial and parametric variables, and hierarchical tensor decompositions separating all variables. We describe corresponding adaptive algorithms based on a common generic template and show their near-optimality with respect to natural approximability assumptions for each type of approximation. A central ingredient in the resulting bounds for the total computational complexity are new operator compression results for the case of infinitely many parameters. We conclude with a comparison of the complexity estimates based on the actual approximability properties of classes of parametric model problems, which shows that the computational costs of optimized low-rank expansions can be significantly lower or higher than those of sparse polynomial expansions, depending on the particular type of parametric problem.

Andrea Bonito, Albert Cohen, Wolfgang Dahmen et al.

We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, Hölder or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.

Markus Bachmayr, Albert Cohen, Giovanni Migliorati

We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments.

Markus Bachmayr, Albert Cohen

Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $n$-widths.

Albert Cohen, Giovanni Migliorati

The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of dimensionality. As a typical example, standard polynomial spaces, such as those of total degree type, are often uneffective to reach a prescribed accuracy unless a prohibitive number of evaluations is invested. In recent years it has been shown that, for certain relevant applications, there are substantial advantages in using certain sparse polynomial spaces having anisotropic features with respect to the different variables. These applications include in particular the numerical approximation of high-dimensional parametric and stochastic partial differential equations. We start by surveying several results in this direction, with an emphasis on the numerical algorithms that are available for the construction of the approximation, in particular through interpolation or discrete least-squares fitting. All such algorithms rely on the assumption that the set of multi-indices associated with the polynomial space is downward closed. In the present paper we introduce some tools for the study of approximation in multivariate spaces under this assumption, and use them in the derivation of error bounds, sometimes independent of the dimension d, and in the development of adaptive strategies.

Markus Bachmayr, Albert Cohen, Giovanni Migliorati

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence $y=(y_j)_{j\geq 1}$ of scalar random variables. One may then apply high-dimensional approximation methods to the solution map $y\mapsto u(y)$. Although Karhunen-Loève representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that they do not generally yield optimal approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen-Loève representations. Our construction is based on a periodic extension of the random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matérn covariances.

Albert Cohen, Nira Dyn, Frédéric Hecht et al.

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).

Jean-Marie Mirebeau, Albert Cohen

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.

Benjamin Arras, Markus Bachmayr, Albert Cohen

We consider the problem of approximating an unknown function $u\in L^2(D,ρ)$ from its evaluations at given sampling points $x^1,\dots,x^n\in D$, where $D\subset \mathbb{R}^d$ is a general domain and $ρ$ is a probability measure. The approximation is picked in a linear space $V_m$ where $m=\dim(V_m)$ and computed by a weighted least squares method. Recent results show the advantages of picking the sampling points at random according to a well-chosen probability measure $μ$ that depends both on $V_m$ and $ρ$. With such a random design, the weighted least squares approximation is proved to be stable with high probability, and having precision comparable to that of the exact $L^2(D,ρ)$-orthonormal projection onto $V_m$, in a near-linear sampling regime $n\sim{m\log m}$. The present paper is motivated by the adaptive approximation context, in which one typically generates a nested sequence of spaces $(V_m)_{m\geq1}$ with increasing dimension. Although the measure $μ=μ_m$ changes with $V_m$, it is possible to recycle the previously generated samples by interpreting $μ_m$ as a mixture between $μ_{m-1}$ and an update measure $σ_m$. Based on this observation, we discuss sequential sampling algorithms that maintain the stability and approximation properties uniformly over all spaces $V_m$. Our main result is that the total number of computed sample at step $m$ remains of the order $m\log{m}$ with high probability. Numerical experiments confirm this analysis.

Gilles Chardon, Albert Cohen, Laurent Daudet

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain $Ω$ from their values at scattered points $x_1,\dots,x_n\subset Ω$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $Ω$ contained in a larger domain $D$ in which the acoustic field propagates. In many applied settings, the shape of $D$ and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to $u$ using these families of functions based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace, the sample size $n$ and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of $Ω$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.

S. VenkataKeerthy, Siddharth Jain, Anilava Kundu et al.

We aim to automate decades of research and experience in register allocation, leveraging machine learning. We tackle this problem by embedding a multi-agent reinforcement learning algorithm within LLVM, training it with the state of the art techniques. We formalize the constraints that precisely define the problem for a given instruction-set architecture, while ensuring that the generated code preserves semantic correctness. We also develop a gRPC based framework providing a modular and efficient compiler interface for training and inference. Our approach is architecture independent: we show experimental results targeting Intel x86 and ARM AArch64. Our results match or out-perform the heavily tuned, production-grade register allocators of LLVM.

Albert Cohen, Giovanni Migliorati, Fabio Nobile

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_Λ:={\rm span} \{y\mapsto y^ν\,: \, ν\in Λ\}$ with $Λ\subset \mathbb{N}_0^d$ over the domain $Γ:=[-1,1]^d$, based on the sampling of this function at points $y^1,\dots,y^m \in Γ$. The samples are independently drawn according to a given probability density $ρ$ belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. We restrict our attention to polynomial spaces associated with \emph{downward closed} sets $Λ$ of \emph{prescribed} cardinality $n$, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in $L^2(Γ,dρ)$ with the best achievable polynomial approximation error when using downward closed sets of cardinality $n$. We establish conditions between the dimension $n$ and the size $m$ of the sample, under which these two errors are proven to be comparable. Our main finding is that the dimension $d$ enters only moderately in the resulting trade-off between $m$ and $n$, in terms of a logarithmic factor $\ln(d)$, and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named {\it anchored} sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon [2] which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs.

Jean-Marie Mirebeau, Albert Cohen

We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.

Albert Cohen, Jean-Marie Mirebeau

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the Lp-norms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction.

S. VenkataKeerthy, Siddharth Jain, Umesh Kalvakuntla et al.

There is a growing interest in enhancing compiler optimizations with ML models, yet interactions between compilers and ML frameworks remain challenging. Some optimizations require tightly coupled models and compiler internals,raising issues with modularity, performance and framework independence. Practical deployment and transparency for the end-user are also important concerns. We propose ML-Compiler-Bridge to enable ML model development within a traditional Python framework while making end-to-end integration with an optimizing compiler possible and efficient. We evaluate it on both research and production use cases, for training and inference, over several optimization problems, multiple compilers and its versions, and gym infrastructures.

Dumitru Potop Butucaru, Albert Cohen, Gordon Plotkin et al.

Reactive languages are dedicated to the programming of systems which interact continuously and concurrently with their environment. Values take the form of unbounded streams modeling the (discrete) passing of time or the sequence of concurrent interactions. While conventional reactivity models recurrences forward in time, we introduce a symmetric reactive construct enabling backward recurrences. Constraints on the latter allow to make the implementation practical. Machine Learning (ML) systems provide numerous motivations for all of this: we demonstrate that reverse-mode automatic differentiation, backpropagation, batch normalization, bidirectional recurrent neural networks, training and reinforcement learning algorithms, are all naturally captured as bidirectional reactive programs.

Milan Bradonjić, Matthew Causley, Albert Cohen

In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller's reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from (16) and used effectively in the Stochastic Optimal Control of Advertising (12). This model of reputation is combined with the effect of online reputation on sales price empirically verified in (9). We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller's reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.

Son Tuan Vu, Albert Cohen, Karine Heydemann et al.

Secure applications implement software protections against side-channel and physical attacks. Such protections are meaningful at machine code or micro-architectural level, but they typically do not carry observable semantics at source level. To prevent optimizing compilers from altering the protection, security engineers embed input/output side-effects into the protection. These side-effects are error-prone and compiler-dependent, and the current practice involves analyzing the generated machine code to make sure security or privacy properties are still enforced. Vu et al. recently demonstrated how to automate the insertion of volatile side-effects in a compiler [52], but these may be too expensive in fined-grained protections such as control-flow integrity. We introduce observations of the program state that are intrinsic to the correct execution of security protections, along with means to specify and preserve observations across the compilation flow. Such observations complement the traditional input/output-preservation contract of compilers. We show how to guarantee their preservation without modifying compilation passes and with as little performance impact as possible. We validate our approach on a range of benchmarks, expressing the secure compilation of these applications in terms of observations to be made at specific program points.

Chris Lattner, Mehdi Amini, Uday Bondhugula et al.

This work presents MLIR, a novel approach to building reusable and extensible compiler infrastructure. MLIR aims to address software fragmentation, improve compilation for heterogeneous hardware, significantly reduce the cost of building domain specific compilers, and aid in connecting existing compilers together. MLIR facilitates the design and implementation of code generators, translators and optimizers at different levels of abstraction and also across application domains, hardware targets and execution environments. The contribution of this work includes (1) discussion of MLIR as a research artifact, built for extension and evolution, and identifying the challenges and opportunities posed by this novel design point in design, semantics, optimization specification, system, and engineering. (2) evaluation of MLIR as a generalized infrastructure that reduces the cost of building compilers-describing diverse use-cases to show research and educational opportunities for future programming languages, compilers, execution environments, and computer architecture. The paper also presents the rationale for MLIR, its original design principles, structures and semantics.

Julien Proy, Karine Heydemann, Fabien Majéric et al.

In the area of physical attacks, system-on-chip (SoC) designs have not received the same level of attention as simpler micro-controllers. We try to model the behavior of secure software running on a superscalar out-of-order microprocessor typical of more complex SoC, in the presence of electromagnetic (EM) pulses. We first show that it is possible, in a black box approach, to corrupt the loop iteration count of both original and hardened versions of two sensitive loops. We propose a characterization methodology based on very simple codes, to understand and classify the fault effects at the level of the instruction set architecture (ISA). The resulting classification includes the well established instruction skip and register corruption models, as well as new effects specific to more complex processors, such as operand substitution, multiple correlated register corruptions, advanced control-flow hijacking, and combinations of all reported effects. This diversity and complexity of effects can lead to powerful attacks. The proposed methodology and fault classification at ISA level is a first step towards a more complete characterization. It is also a tool supporting the designers of software and hardware countermeasures.

Nicolas Vasilache, Oleksandr Zinenko, Theodoros Theodoridis et al.

Deep learning models with convolutional and recurrent networks are now ubiquitous and analyze massive amounts of audio, image, video, text and graph data, with applications in automatic translation, speech-to-text, scene understanding, ranking user preferences, ad placement, etc. Competing frameworks for building these networks such as TensorFlow, Chainer, CNTK, Torch/PyTorch, Caffe1/2, MXNet and Theano, explore different tradeoffs between usability and expressiveness, research or production orientation and supported hardware. They operate on a DAG of computational operators, wrapping high-performance libraries such as CUDNN (for NVIDIA GPUs) or NNPACK (for various CPUs), and automate memory allocation, synchronization, distribution. Custom operators are needed where the computation does not fit existing high-performance library calls, usually at a high engineering cost. This is frequently required when new operators are invented by researchers: such operators suffer a severe performance penalty, which limits the pace of innovation. Furthermore, even if there is an existing runtime call these frameworks can use, it often doesn't offer optimal performance for a user's particular network architecture and dataset, missing optimizations between operators as well as optimizations that can be done knowing the size and shape of data. Our contributions include (1) a language close to the mathematics of deep learning called Tensor Comprehensions, (2) a polyhedral Just-In-Time compiler to convert a mathematical description of a deep learning DAG into a CUDA kernel with delegated memory management and synchronization, also providing optimizations such as operator fusion and specialization for specific sizes, (3) a compilation cache populated by an autotuner. [Abstract cutoff]

Markus Bachmayr, Albert Cohen, Dinh Dũng et al.

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in $\ell^p$ summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for non-Hilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.

Albert Cohen, Giovanni Migliorati

We consider the problem of reconstructing an unknown bounded function $u$ defined on a domain $X\subset \mathbb{R}^d$ from noiseless or noisy samples of $u$ at $n$ points $(x^i)_{i=1,\dots,n}$. We measure the reconstruction error in a norm $L^2(X,dρ)$ for some given probability measure $dρ$. Given a linear space $V_m$ with ${\rm dim}(V_m)=m\leq n$, we study in general terms the weighted least-squares approximations from the spaces $V_m$ based on independent random samples. The contribution of the present paper is twofold. From the theoretical perspective, we establish results in expectation and in probability for weighted least squares in general approximation spaces $V_m$. These results show that for an optimal choice of sampling measure $dμ$ and weight $w$, which depends on the space $V_m$ and on the measure $dρ$, stability and optimal accuracy are achieved under the mild condition that $n$ scales linearly with $m$ up to an additional logarithmic factor. The present analysis covers also cases where the function $u$ and its approximants from $V_m$ are unbounded, which might occur for instance in the relevant case where $X=\mathbb{R}^d$ and $dρ$ is the Gaussian measure. From the numerical perspective, we propose a sampling method which allows one to generate independent and identically distributed samples from the optimal measure $dμ$. This method becomes of interest in the multivariate setting where $dμ$ is generally not of tensor product type. We illustrate this for particular examples of approximation spaces $V_m$ of polynomial type, where the domain $X$ is allowed to be unbounded and high or even infinite dimensional, motivated by certain applications to parametric and stochastic PDEs.

Markus Bachmayr, Albert Cohen, Ronald DeVore et al.

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Loève representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion.

Peter Binev, Albert Cohen, Wolfgang Dahmen et al.

We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from $m$ measurements obtained through known linear functionals on $\mathcal{H}$. Problems of this type are well studied \cite{MRW} under an assumption that $u$ belongs to a prescribed model class, e.g. a known compact subset of $\mathcal{H}$. Motivated by reduced modeling for parametric partial differential equations, this paper considers another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$, or more generally, how well $u$ can be approximated by each $k$-dimensional subspace $V_k$ of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$. A recovery algorithm for the one-space formulation, proposed in \cite{MPPY}, is proven here to be optimal and to have a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multi-space case for which it is shown that the set of all $u$ satisfying the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is to identify any point in this intersection which can provide a much better accuracy than in the one-space problem. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem. A detailed analysis of one of them provides a posteriori performance estimates for the iterates, stopping criteria, and convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.

Albert Cohen, Wolfgang Dahmen, Ronald DeVore

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary $\mathcal{D}$ in a Hilbert space $\mathcal{H}$. Given an element $f\in \mathcal{H}$, OMP generates a sequence of approximations $f_n$, $n=1,2,\dots$, each of which is a linear combination of $n$ dictionary elements chosen by a greedy criterion. It is studied whether the approximations $f_n$ are in some sense comparable to {\em best $n$ term approximation} from the dictionary. One important result related to this question is a theorem of Zhang \cite{TZ} in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers $n$-sparse signal, whenever the dictionary $\mathcal{D}$ satisfies a Restricted Isometry Property (RIP) of order $An$ for some constant $A$, and that the procedure is also stable in $\ell^2$ under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang's theorem, formulated in the general context of $n$ term approximation from a dictionary in arbitrary Hilbert spaces $\mathcal{H}$. Namely, it is shown that OMP generates near best $n$ term approximations under a similar RIP condition.

Andi Drebes, Karine Heydemann, Antoniu Pop et al.

To efficiently exploit the resources of new many-core architectures, integrating dozens or even hundreds of cores per chip, parallel programming models have evolved to expose massive amounts of parallelism, often in the form of fine-grained tasks. Task-parallel languages, such as OpenStream, X10, Habanero Java and C or StarSs, simplify the development of applications for new architectures, but tuning task-parallel applications remains a major challenge. Performance bottlenecks can occur at any level of the implementation, from the algorithmic level (e.g., lack of parallelism or over-synchronization), to interactions with the operating and runtime systems (e.g., data placement on NUMA architectures), to inefficient use of the hardware (e.g., frequent cache misses or misaligned memory accesses); detecting such issues and determining the exact cause is a difficult task. In previous work, we developed Aftermath, an interactive tool for trace-based performance analysis and debugging of task-parallel programs and run-time systems. In contrast to other trace-based analysis tools, such as Paraver or Vampir, Aftermath offers native support for tasks, i.e., visualization, statistics and analysis tools adapted for performance debugging at task granularity. However, the tool currently does not provide support for the automatic detection of performance bottlenecks and it is up to the user to investigate the relevant aspects of program execution by focusing the inspection on specific slices of a trace file. In this paper, we present ongoing work on two extensions that guide the user through this process.