Jump-sparse and sparse recovery using Potts functionals

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $\ell^1$ minimization (sparse signals)

Parity space-based fault detection for Markovian jump systems

This article deals with problems of parity space-based fault detection for a class of discrete-time linear Markovian jump systems. A new algorithm is firstly introduced to reduce the computation of mode-dependent redundancy relation parameter matrices. Different from the case of linear time invariant systems, the parity space-based residual generator for a Markovian jump system cannot be designed off-line because it depends on the history of system modes in the last finite steps. In order to overcome this difficulty, a finite set of parity matrices is pre-designed applying a unified approach to linear time invariant systems. Then the on-line residual generation can be easily implemented. Moreover, the problem of residual evaluation is also considered which includes the determination of a residual evaluation function and a threshold. Finally, a numerical example is given to illustrate the effectiveness of the proposed method

Joint Image Reconstruction and Segmentation Using the Potts Model

We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from $7$ angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data

A Unified Analysis of Stochastic Optimization Methods Using Jump System Theory and Quadratic Constraints

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).Comment: To Appear in Proceedings of the Annual Conference on Learning Theory (COLT) 201

H ∞ Model Reduction of 2D Markovian Jump System with Roesser Model

Abstract: This paper extends the results obtained for one-dimensional Markovian jump systems to investigate the problem of H ∞ model reduction for a class of linear discrete time 2D Markovian jump systems with state delays in Roesser model which is time-varying and mode-independent. The reduced-order model with the same randomly jumping parameters is proposed which can make the error systems stochastically stable with a prescribed H ∞ performance. A sufficient condition in terms of linear matrix inequalities (LMIs) plus matrix inverse constraints are derived for the existence of a solution to the reduced-order model problems. The cone complimentarity linearization (CCL) method is exploited to cast them into nonlinear minimization problems subject to LMI constraints. A numerical example is given to illustrate the design procedures

Drift dependence of optimal trade execution strategies under transient price impact

We give a complete solution to the problem of minimizing the expected liquidity costs in presence of a general drift when the underlying market impact model has linear transient price impact with exponential resilience. It turns out that this problem is well-posed only if the drift is absolutely continuous. Optimal strategies often do not exist, and when they do, they depend strongly on the derivative of the drift. Our approach uses elements from singular stochastic control, even though the problem is essentially non-Markovian due to the transience of price impact and the lack in Markovian structure of the underlying price process. As a corollary, we give a complete solution to the minimization of a certain cost-risk criterion in our setting