Matrix probing and its conditioning

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media

Matrix probing, skeleton decompositions, and sparse Fourier transform

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 163-168).In this thesis, we present three different randomized algorithms that help to solve matrices, compute low rank approximations and perform the Fast Fourier Transform. Matrix probing and its conditioning When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B1,... , Bp, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to obtain an approximation to A-1. A basic question is whether this linear system is well-conditioned. This is important for two reasons: a well-conditioned system means (1) we can invert it and (2) the error in the reconstruction can be controlled. In this paper, we show that if the Gram matrix of the Bj's is sufficiently well-conditioned and each Bj has a high numerical rank, then n [alpha] p log2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We also demonstrate numerically that matrix probing can produce good preconditioners for inverting elliptic operators in variable media. Skeleton decompositions in sublinear time A skeleton decomposition of a matrix A is any factorization of the form A:CZAR: where A:C comprises columns of A, and AR: comprises rows of A. In this paper, we investigate the conditions under which random sampling of C and R results in accurate skeleton decompositions. When the singular vectors (or more generally the generating vectors) are incoherent, we show that a simple algorithm returns an accurate skeleton in sublinear O(l3) time from l ~/- k logn rows and columns drawn uniformly at random, with an approximation error of the form O(n/l[sigma]k) where 0k is the k-th singular value of A. We discuss the crucial role that regularization plays in forming the middle matrix U as a pseudo-inverse of the restriction ARC of A to rows in R and columns in C. The proof methods enable the analysis of two alternative sublinear-time algorithms, based on the rank-revealing QR decomposition, which allow us to tighten the number of rows and/or columns sampled to k with an error bound proportional to [sigma]-k. Sparse Fourier transform using the matrix pencil method One of the major applications of the FFT is to compress frequency-sparse signals. Yet, FFT algorithms do not leverage on this sparsity. Say we want to perform the Fourier transform on [epsilon] E CN to obtain some [chi], which is known to be S-sparse with some additive noise. Even when S is small, FFT still takes O(N log N) time. In contrast, SFT (sparse Fourier transform) algorithms aim to run in Õ(S)-time ignoring log factors. Unfortunately, SFT algorithms are not widely used because they are faster than the FFT only when S << N. We hope to address this deficiency. In this work, we present the fastest known robust Õ(S)-time algorithm which can run up to 20 times faster than the current state-of-the-art algorithm AAFFT. The major new ingredient is a mode collision detector using the matrix pencil method. This enables us to do away with a time-consuming coefficient estimation loop, use a cheaper filter and take fewer samples of x. We also speed up a crucial basic operation of many SFT algorithms by halving the number of trigonometric computations. Our theory is however not complete. First, we prove that the collision detector works for a few classes of random signals. Second, we idealize the behavior of the collision detector and show that with good probability, our algorithm runs in O(S log 2 - log N) time and outputs a O(S)-sparse [chi]' such that [mathematical formula inserted] where [chi], is the best exact S-sparse approximation of [chi].by Jiawei Chiu.Ph.D

Matrix probing: a randomized preconditioner for the wave-equation Hessian

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.Comment: 21 pages, 6 figure

Towards Full Automated Drive in Urban Environments: A Demonstration in GoMentum Station, California

Each year, millions of motor vehicle traffic accidents all over the world cause a large number of fatalities, injuries and significant material loss. Automated Driving (AD) has potential to drastically reduce such accidents. In this work, we focus on the technical challenges that arise from AD in urban environments. We present the overall architecture of an AD system and describe in detail the perception and planning modules. The AD system, built on a modified Acura RLX, was demonstrated in a course in GoMentum Station in California. We demonstrated autonomous handling of 4 scenarios: traffic lights, cross-traffic at intersections, construction zones and pedestrians. The AD vehicle displayed safe behavior and performed consistently in repeated demonstrations with slight variations in conditions. Overall, we completed 44 runs, encompassing 110km of automated driving with only 3 cases where the driver intervened the control of the vehicle, mostly due to error in GPS positioning. Our demonstration showed that robust and consistent behavior in urban scenarios is possible, yet more investigation is necessary for full scale roll-out on public roads.Comment: Accepted to Intelligent Vehicles Conference (IV 2017

Multiscale characterization of the 3D network structure of metal carbides in a Ni superalloy by synchrotron X-ray microtomography and ptychography

Synchrotron X-ray microtomography and ptychography were used to characterize the 3D network structure, morphology and distribution of metal carbides in an as-cast IN713LC Ni superalloy. MC typed carbides were found to distribute mainly on the grain boundary between the matrix γ and γ' phase. The differences in solidification cooling rate had a minor influence on the volume fraction of the MC type carbides, but significantly affected the carbide size, distribution and network morphology. Depending on the local composition of the remaining liquid phase and geometric constraints, the carbides can form either spherical or strip or network morphologies. The research demonstrated clearly the advantage and technical potential of using the two complementary tomography techniques synergistically to characterize non-destructively complex multiple-phase structures in three dimensional space with a spatial resolution of ~30 nm

A Cluster-based Personalized Item Recommended Approach on the Educational Assessment System

Personalized item recommendation enables the educational assessment system to make deliberate efforts to perform appropriate assessment strategies that fit the needs, purposes, preferences, and interests of individual teachers. This study presents a dynamically personalized item-recommendation approach that is based on clustering in-serve teachers with assessment compiling interest and preference characteristics to recommend available, best-fit candidate items to support teachers to construct their classroom assessment. A two-round assessment constructing activity was being adopted to collect and extract these teacher’ assessment knowledge (item selected preference behaviors), and through the designed item-recommendation mechanism to facilitate IKMAAS [1] to recommend proper items to meet different individual in-serve teachers. To evaluate the effectiveness and usability for the cluster-based personalized item-recommendation, the assessment system log analysis and the questionnaire collected from participating teachers’ perceptions were being used. The results showed the proposed item-recommendation approach based on clustered teachers’ assessment knowledge can effectively improve their educational assessment construction

Sublinear Randomized Algorithms for Skeleton Decompositions

A skeleton decomposition of a matrix A is any factorization of the form A[subscript :C]ZA[subscript R:], where A[subscript :C] comprises columns of A, and A[subscript R:] comprises rows of A. In this paper, we investigate the conditions under which random sampling of C and R results in accurate skeleton decompositions. When the singular vectors (or more generally the generating vectors) are incoherent, we show that a simple algorithm returns an accurate skeleton in sublinear O(ℓ[superscript 3]) time from ℓ ~ k log n rows and columns drawn uniformly at random, with an approximation error of the form O([n over ℓ]σ[subscript k]) whereσ[subscript k] is the kth singular value of A. We discuss the crucial role that regularization plays in forming the middle matrix U as a pseudoinverse of the restriction A[subscript RC] of A to rows in R and columns in C. The proof methods enable the analysis of two alternative sublinear-time algorithms, based on the rank-revealing QR decomposition, which allows us to tighten the number of rows and/or columns to k with error bound proportional to σ[subscript k].National Science Foundation (U.S.)Alfred P. Sloan Foundatio

A wearable freestanding electrochemical sensing system.

To render high-fidelity wearable biomarker data, understanding and engineering the information delivery pathway from epidermally retrieved biofluid to a readout unit are critical. By examining the biomarker information delivery pathway and recognizing near-zero strained regions within a microfluidic device, a strain-isolated pathway to preserve biomarker data fidelity is engineered. Accordingly, a generalizable and disposable freestanding electrochemical sensing system (FESS) is devised, which simultaneously facilitates sensing and out-of-plane signal interconnection with the aid of double-sided adhesion. The FESS serves as a foundation to realize a system-level design strategy, addressing the challenges of wearable biosensing, in the presence of motion, and integration with consumer electronics. To this end, a FESS-enabled smartwatch was developed, featuring sweat sampling, electrochemical sensing, and data display/transmission, all within a self-contained wearable platform. The FESS-enabled smartwatch was used to monitor the sweat metabolite profiles of individuals in sedentary and high-intensity exercise settings