In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set {f_i} of polynomials over a ring R, the solution set is the subset of R on which the polynomials all vanish (evaluate to 0), formally : {x\in R: \forall i\in I, f_i(x)=0} The feasible region of a constrained optimization problem is the solution set of the constraints. Examples

The solution set of the single equation x=0 is the set {0}.

For any non-zero polynomial f over the complex numbers in one variable, the solution set is made up of finitely many points.

However, for a complex polynomial in more than one variable the solution set has no isolated points.

Remarks In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets. Other meanings More generally, the solution set to

Sparsest piecewise-linear regression of one-dimensional data

Thomas Jean Debarre, Quentin Alain Denoyelle, Julien René Fageot, Michaël Unser

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty with TV regularization is that the solution is generally non-unique, an aspect that is often ignored in practice. In this paper, we present a systematic analysis that results in a complete description of the solution set with a clear distinction between the cases where the solution is unique and those, much more frequent, where it is not. For the latter scenario, we identify the sparsest solutions, i.e., those with the minimum number of knots, and we derive a formula to compute the minimum number of knots based solely on the data points. To achieve this, we first consider the problem of exact interpolation which leads to an easier theoretical analysis. Next, we relax the exact interpolation requirement to a regression setting, and we consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. Based on our theoretical analysis, we propose a simple and fast two-step algorithm, agnostic to uniqueness, to reach a sparsest solution of this penalized problem.

Elsevier2021

On canonical representation of convex polyhedra

Stefano Picozzi

Convex polyhedra are important objects in various areas of mathematics and other disciplines. A fundamental result, known as Minkowski-Weyl theorem, states that every polyhedron admits two types of representations, either as the solution set to a finite system of linear inequalities or the Minkowski sum of a finite set of points and half-rays. These are usually referred to as its H-representations and its V-representations, respectively. Neither H-representations nor V-representations are unique. Hence, deciding whether two H-representations, or two V-representations describe the same polyhedron is a nontrivial problem. We identify particular representations which are easy to determine, are compact, and reflect the geometrical properties of the underlying polyhedron. Key ingredients to this discussion are affine transformations of polyhedra, duality and complexity of polyhedral decision problems. In this dissertation, we discuss in detail the problem of refining the definitions of Hand V- representations such as to guarantee a one to one correspondence between polyhedra and their representations. As a convenient formalism, we introduce the notion of the canonical representation rule, which is any function assigning to each polyhedron P a particular H-representation, and a particular V-representation. The orthogonal representation rule is a well-known example of such a function. We define the lexico-smallest representation rule, an improved canonical representation rule that produce representations which are nonredundant, make the underlying geometry transparent and, furthermore, are sparse. These rules also exhibit nice polarity properties that can be exploited for computation. The key computational tool is linear programing. Furthermore, we show that the lexico-smallest H-representation of a perfect bipartite matching polyhedron is easy to determine using the combinatorial properties of the underlying graph. Finally, we discuss the complexity of various polyhedral verification problems related to representation of convex polyhedra. The standard technique to identify the redundancies within an H- or a V-representation is to use linear programming. We discuss the complexity of the converse reduction. More precisely, we show that deciding the feasibility of a system of linear inequalities can be reduced to deciding redundancy in an H- or in a V- representation in linear time. Also, we will study the equivalence of these problems with those of identifying the implicit linearities within an H- or a V- representation, and the boundedness of a polyhedron.

EPFL2005

From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.

Harshit Gupta

In this thesis, we propose new algorithms to solve inverse problems in the context of biomedical images. Due to ill-posedness, solving these problems require some prior knowledge of the statistics of the underlying images. The traditional algorithms, in the field, assume prior knowledge related to smoothness or sparsity of these images. Recently, they have been outperformed by the second generation algorithms which harness the power of neural networks to learn required statistics from training data. Even more recently, last generation deep-learning-based methods have emerged which require neither training nor training data. This thesis devises algorithms which progress through these generations. It extends these generations to novel formulations and applications while bringing more robustness. In parallel, it also progresses in terms of complexity, from proposing algorithms for problems with 1D data and an exact known forward model to the ones with 4D data and an unknown parametric forward model. We introduce five main contributions. The last three of them propose deep-learning-based latest-generation algorithms that require no prior training. 1) We develop algorithms to solve the continuous-domain formulation of inverse problems with both classical Tikhonov and total-variation regularizations. We formalize the problems, characterize the solution set, and devise numerical approaches to find the solutions. 2) We propose an algorithm that improves upon end-to-end neural-network-based second generation algorithms. In our method, a neural network is first trained as a projector on a training set, and is then plugged in as a projector inside the projected gradient descent (PGD). Since the problem is nonconvex, we relax the PGD to ensure convergence to a local minimum under some constraints. This method outperforms all the previous generation algorithms for Computed Tomography (CT). 3) We develop a novel time-dependent deep-image-prior algorithm for modalities that involve a temporal sequence of images. We parameterize them as the output of an untrained neural network fed with a sequence of latent variables. To impose temporal directionality, the latent variables are assumed to lie on a 1D manifold. The network is then tuned to minimize the data fidelity. We obtain state-of-the-art results in dynamic magnetic resonance imaging (MRI) and even recover intra-frame images. 4) We propose a novel reconstruction paradigm for cryo-electron-microscopy (CryoEM) called CryoGAN. Motivated by generative adversarial networks (GANs), we reconstruct a biomolecule's 3D structure such that its CryoEM measurements resemble the acquired data in a distributional sense. The algorithm is pose-or-likelihood-estimation-free, needs no ab initio, and is proven to have a theoretical guarantee of recovery of the true structure. 5) We extend CryoGAN to reconstruct continuously varying conformations of a structure from heterogeneous data. We parameterize the conformations as the output of a neural network fed with latent variables on a low-dimensional manifold. The method is shown to recover continuous protein conformations and their energy landscape.

EPFL2020

On canonical representation of convex polyhedra

Stefano Picozzi

EPFL2005

From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.

Harshit Gupta

EPFL2020

Solution set

The solution set of the single equation x=0 is the set {0}.

For any non-zero polynomial f over the complex numbers in one variable, the solution set is made up of finitely many points.

However, for a complex polynomial in more than one variable the solution set has no isolated points.

Sparsest piecewise-linear regression of one-dimensional data

On canonical representation of convex polyhedra

From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.

Sparsest piecewise-linear regression of one-dimensional data

On canonical representation of convex polyhedra

From Classical to Unsupervised-Deep-Learning Methods for Solving Inverse Problems in Imaging.