Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Recipes for sparse LDA of horizontal data

Many important modern applications require analyzing data with more variables than observations, called for short horizontal. In such situation the classical Fisher’s linear discriminant analysis (LDA) does not possess solution because the within-group scatter matrix is singular. Moreover, the number of the variables is usually huge and the classical type of solutions (discriminant functions) are difficult to interpret as they involve all available variables. Nowadays, the aim is to develop fast and reliable algorithms for sparse LDA of horizontal data. The resulting discriminant functions depend on very few original variables, which facilitates their interpretation. The main theoretical and numerical challenge is how to cope with the singularity of the within-group scatter matrix. This work aims at classifying the existing approaches according to the way they tackle this singularity issue, and suggest new ones

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Benign landscapes of low-dimensional relaxations for orthogonal synchronization on general graphs

Orthogonal group synchronization is the problem of estimating $n$ elements $Z_1, \ldots, Z_n$ from the orthogonal group $\mathrm{O}(r)$ given some relative measurements $R_{ij} \approx Z_i^{}Z_j^{-1}$. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from $O(n)$ to $O(n^2)$. Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension $O(n^{3/2})$. For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that nonconvex relaxations of only slightly increased dimension seem sufficient for SLAM problems. We partially explain this. This also has implications for Kuramoto oscillators. Specifically, we minimize the least-squares cost function in terms of estimators $Y_1, \ldots, Y_n$. Each $Y_i$ is relaxed to the Stiefel manifold $\mathrm{St}(r, p)$ of $r \times p$ matrices with orthonormal rows. The available measurements implicitly define a (connected) graph $G$ on $n$ vertices. In the noiseless case, we show that second-order critical points are globally optimal as soon as $p \geq r+2$ for all connected graphs $G$. (This implies that Kuramoto oscillators on $\mathrm{St}(r, p)$ synchronize for all $p \geq r + 2$.) This result is the best possible for general graphs; the previous best known result requires $2p \geq 3(r + 1)$. For $p > r + 2$, our result is robust to modest amounts of noise (depending on $p$ and $G$). When local minima remain, they still achieve minimax-optimal error rates. Our proof uses a novel randomized choice of tangent direction to prove (near-)optimality of second-order critical points. Finally, we partially extend our noiseless landscape results to the complex case (unitary group), showing that there are no spurious local minima when $2p \geq 3r$

Sparse PCA and investigation of multielements compositional repositories: theory and applications

The geochemistry of floodplain sediments is fundamental to monitor environmental changes and to quantify their contribution to natural and anthropic processes. A floodplain sediment composition is a vector of positive elements which sum to a fixed constant. The analysis of high-dimensional compositions requires methods that produce results involving only a small portion of the original variables. On the other hand, the analysis must take into account the additional constraints specific to compositions. With the purpose of studying these problems, a new procedure for sparse PCA is proposed on European floodplain sediment samples

Predictive value of admission hyperglycaemia on mortality in patients with acute myocardial infarction.

International audienceRATIONALE AND AIM: In patients with an acute myocardial infarction, admission hyperglycaemia (AH) is a major risk factor for mortality. However, the predictive value of AH, when the risk score and use of guidelines-recommended treatments are considered, is poorly documented. METHODS: The first fasting plasma glucose levels after admission, risk level, guidelines-recommended treatment use and 1-year mortality were recorded. Patients with first fasting glucose level after admission > 7.7 mmo/l were considered to have AH. RESULTS: Three hundred and twenty patients with ST segment elevation myocardial infarction (STEMI) and 404 with non-ST segment elevation myocardial infarction (NSTEMI) were included. One hundred and seventy-five (24%) patients had pre-existing diabetes (diabetes group), 154 (21%) had AH (AH+ group) and the remainding 395 (55%) had neither diabetes nor AH (AH- group). The Global Registry of Acute Coronary Events (GRACE) risk score was lower in the AH- group, but the use of guidelines-recommended treatment was comparable in all groups. At 1 year, the mortality rate was higher in the AH+ group compared with the AH- group (18.8 vs. 6.1%, P < 0.01) and similar to that in the diabetes group (18.8 vs. 16.6%, P = NS). The relation between glycaemic status and mortality remained strong [AH+ vs. AH-, OR = 3.0 (1.5, 6.0) and diabetes vs. AH-, OR = 3.6 (1.7, 6.6)] after adjustment for the GRACE risk score [OR = 2.4 (1.8, 3.1) per 10% increase] and for treatment score [OR = 0.7 (0.6, 0.8) per 10% increase]. CONCLUSIONS: In patients without a history of diabetes, the presence of AH indicates an increased risk of 1-year mortality, similar to that of patients with diabetes, even when the risk score and use of guidelines-recommended treatment are controlled for