Sparse Recovery via Differential Inclusions

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to \emph{Linearized Bregman iteration}, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal $l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.Comment: In Applied and Computational Harmonic Analysis, 201

A Consistent Histogram Estimator for Exchangeable Graph Models

Exchangeable graph models (ExGM) subsume a number of popular network models. The mathematical object that characterizes an ExGM is termed a graphon. Finding scalable estimators of graphons, provably consistent, remains an open issue. In this paper, we propose a histogram estimator of a graphon that is provably consistent and numerically efficient. The proposed estimator is based on a sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree of a graph, then smooths the sorted graph using total variation minimization. The consistency of the SAS algorithm is proved by leveraging sparsity concepts from compressed sensing.Comment: 28 pages, 5 figure

On the monotone and primal-dual active set schemes for ℓp\ell^p-type problems, p∈(0,1]p \in (0,1]

Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm, $p \in (0, 1]$, of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction

Minimizing L1 over L2 norms on the gradient

In this paper, we study the L1/L2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L1/L2 is better than the L1 norm when approximating the L0 norm to promote sparsity. Consequently, we postulate that applying L1/L2 on the gradient is better than the classic total variation (the L1 norm on the gradient) to enforce the sparsity of the image gradient. To verify our hypothesis, we consider a constrained formulation to reveal empirical evidence on the superiority of L1/L2 over L1 when recovering piecewise constant signals from low-frequency measurements. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L1/L2 over L1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of MRI and CT reconstruction. All the numerical results show the efficiency of our proposed approach.Comment: 26 page

Compressive Wave Computation

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology.Comment: 45 pages, 4 figure

Robust strategies for glucose control in type 1 diabetes

[EN] Type 1 diabetes mellitus is a chronic and incurable disease that affects millions of people all around the world. Its main characteristic is the destruction (totally or partially) of the beta cells of the pancreas. These cells are in charge of producing insulin, main hormone implied in the control of blood glucose. Keeping high levels of blood glucose for a long time has negative health effects, causing different kinds of complications. For that reason patients with type 1 diabetes mellitus need to receive insulin in an exogenous way. Since 1921 when insulin was first isolated to be used in humans and first glucose monitoring techniques were developed, many advances have been done in clinical treatment with insulin. Currently 2 main research lines focused on improving the quality of life of diabetic patients are opened. The first one is concentrated on the research of stem cells to replace damaged beta cells and the second one has a more technological orientation. This second line focuses on the development of new insulin analogs to allow emulating with higher fidelity the endogenous pancreas secretion, the development of new noninvasive continuous glucose monitoring systems and insulin pumps capable of administering different insulin profiles and the use of decision-support tools and telemedicine. The most important challenge the scientific community has to overcome is the development of an artificial pancreas, that is, to develop algorithms that allow an automatic control of blood glucose. The main difficulty avoiding a tight glucose control is the high variability found in glucose metabolism. This fact is especially important during meal compensation. This variability, together with the delay in subcutaneous insulin absorption and action causes controller overcorrection that leads to late hypoglycemia (the most important acute complication of insulin treatment). The proposals of this work pay special attention to overcome these difficulties. In that way interval models are used to represent the patient physiology and to be able to take into account parametric uncertainty. This type of strategy has been used in both the open loop proposal for insulin dosage and the closed loop algorithm. Moreover the idea behind the design of this last proposal is to avoid controller overcorrection to minimize hypoglycemia while adding robustness against glucose sensor failures and over/under- estimation of meal carbohydrates. The algorithms proposed have been validated both in simulation and in clinical trials.[ES] La diabetes mellitus tipo 1 es una enfermedad crónica e incurable que afecta a millones de personas en todo el mundo. Se caracteriza por una destrucción total o parcial de las células beta del páncreas. Estas células son las encargadas de producir la insulina, hormona principal en el control de glucosa en sangre. Valores altos de glucosa en la sangre mantenidos en el tiempo afectan negativamente a la salud, provocando complicaciones de diversa índole. Es por eso que los pacientes con diabetes mellitus tipo 1 necesitan recibir insulina de forma exógena. Desde que se consiguiera en 1921 aislar la insulina para poder utilizarla en clínica humana, y se empezaran a desarrollar las primeras técnicas de monitorización de glucemia, se han producido grandes avances en el tratamiento con insulina. Actualmente, las líneas de investigación que se están siguiendo en relación a la mejora de la calidad de vida de los pacientes diabéticos, tienen fundamentalmente 2 vertientes: una primera que se centra en la investigación en células madre para la reposición de las células beta y una segunda vertiente de carácter más tecnológico. Dentro de esta segunda vertiente, están abiertas varias líneas de investigación, entre las que se encuentran el desarrollo de nuevos análogos de insulina que permitan emular más fielmente la secreción endógena del páncreas, el desarrollo de monitores continuos de glucosa no invasivos, bombas de insulina capaces de administrar distintos perfiles de insulina y la inclusión de sistemas de ayuda a la decisión y telemedicina. El mayor reto al que se enfrentan los investigadores es el de conseguir desarrollar un páncreas artificial, es decir, desarrollar algoritmos que permitan disponer de un control automático de la glucosa. La principal barrera que se encuentra para conseguir un control riguroso de la glucosa es la alta variabilidad que presenta su metabolismo. Esto es especialmente significativo durante la compensación de las comidas. Esta variabilidad junto con el retraso en la absorción y actuación de la insulina administrada de forma subcutánea favorece la aparición de hipoglucemias tardías (complicación aguda más importante del tratamiento con insulina) a consecuencia de la sobreactuación del controlador. Las propuestas presentadas en este trabajo hacen especial hincapié en sobrellevar estas dificultades. Así, se utilizan modelos intervalares para representar la fisiología del paciente, y poder tener en cuenta la incertidumbre en sus parámetros. Este tipo de estrategia se ha utilizado tanto en la propuesta de dosificación automática en lazo abierto como en el algoritmo en lazo cerrado. Además la principal idea de diseño de esta última propuesta es evitar la sobreactuación del controlador evitando hipoglucemias y añadiendo robustez ante fallos en el sensor de glucosa y en la estimación de las comidas. Los algoritmos propuestos han sido validados en simulación y en clínica.[CA] La diabetis mellitus tipus 1 és una malaltia crònica i incurable que afecta milions de persones en tot el món. Es caracteritza per una destrucció total o parcial de les cèl.lules beta del pàncrees. Aquestes cèl.lules són les encarregades de produir la insulina, hormona principal en el control de glucosa en sang. Valors alts de glucosa en la sang mantinguts en el temps afecten negativament la salut, provocant complicacions de diversa índole. És per això que els pacients amb diabetis mellitus tipus 1 necessiten rebre insulina de forma exògena. Des que s'aconseguís en 1921 aïllar la insulina per a poder utilitzar-la en clínica humana, i es començaren a desenrotllar les primeres tècniques de monitorització de glucèmia, s'han produït grans avanços en el tractament amb insulina. Actualment, les línies d'investigació que s'estan seguint en relació a la millora de la qualitat de vida dels pacients diabètics, tenen fonamentalment 2 vessants: un primer que es centra en la investigació de cèl.lules mare per a la reposició de les cèl.lules beta i un segon vessant de caràcter més tecnològic. Dins d' aquest segon vessant, estan obertes diverses línies d'investigació, entre les que es troben el desenrotllament de nous anàlegs d'insulina que permeten emular més fidelment la secreció del pàncrees, el desenrotllament de monitors continus de glucosa no invasius, bombes d'insulina capaces d'administrar distints perfils d'insulina i la inclusió de sistemes d'ajuda a la decisió i telemedicina. El major repte al què s'enfronten els investigadors és el d'aconseguir desenrotllar un pàncrees artificial, és a dir, desenrotllar algoritmes que permeten disposar d'un control automàtic de la glucosa. La principal barrera que es troba per a aconseguir un control rigorós de la glucosa és l'alta variabilitat que presenta el seu metabolisme. Açò és especialment significatiu durant la compensació dels menjars. Aquesta variabilitat junt amb el retard en l'absorció i actuació de la insulina administrada de forma subcutània afavorix l'aparició d'hipoglucèmies tardanes (complicació aguda més important del tractament amb insulina) a conseqüència de la sobreactuació del controlador. Les propostes presentades en aquest treball fan especial insistència en suportar aquestes dificultats. Així, s'utilitzen models intervalares per a representar la fisiologia del pacient, i poder tindre en compte la incertesa en els seus paràmetres. Aquest tipus d'estratègia s'ha utilitzat tant en la proposta de dosificació automàtica en llaç obert com en l' algoritme en llaç tancat. A més, la principal idea de disseny d'aquesta última proposta és evitar la sobreactuació del controlador evitant hipoglucèmies i afegint robustesa.Revert Tomás, A. (2015). Robust strategies for glucose control in type 1 diabetes [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56001TESI