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This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.

In optical diffraction tomography (ODT), the three-dimensional scattering potential of a microscopic object rotating around its center is recovered by a series of illuminations with coherent light. Reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane, whereas often only intensities, i.e., phaseless measurements, are available in practice. We propose a new reconstruction approach for ODT with unknown phase information based on three key ingredients. First, the light propagation is modeled using Born's approximation enabling us to use the Fourier diffraction theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier transform via total variation regularization utilizing a primal-dual iteration, which also yields a novel numerical inversion formula for ODT with known phase. The third ingredient is a hybrid input-output scheme. We achieved convincing numerical results, which indicate that ODT with phaseless data is possible. The so-obtained 2D and 3D reconstructions are even comparable to the ones with known phase.

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-$C^1$ are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are $C^1$ smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) $C^1$ smooth across all other edges. Thus, the splines form $H^2$-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-$C^1$ splines are described in an explicit B\'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior.

The Schl\"omilch integral, a generalization of the Dirichlet integral on the simplex, and related probability distributions are reviewed. A distribution that unifies several generalizations of the Dirichlet distribution is presented, with special cases including the scaled Dirichlet distribution and certain Dirichlet mixture distributions. Moments and log-ratio covariances are found, where tractable. The normalization of the distribution motivates a definition, in terms of a simplex integral representation, of complete homogeneous symmetric polynomials of fractional degree.

We consider the problem of recovering a signal from the magnitudes of affine measurements, which is also known as {\em affine phase retrieval}. In this paper, we formulate affine phase retrieval as an optimization problem and develop a second-order algorithm based on Newton method to solve it. Besides being able to convert into a phase retrieval problem, affine phase retrieval has its unique advantages in its solution. For example, the linear information in the observation makes it possible to solve this problem with second-order algorithms under complex measurements. Another advantage is that our algorithm doesn't have any special requirements for the initial point, while an appropriate initial value is essential for most non-convex phase retrieval algorithms. Starting from zero, our algorithm generates iteration point by Newton method, and we prove that the algorithm can quadratically converge to the true signal without any ambiguity for both Gaussian measurements and CDP measurements. In addition, we also use some numerical simulations to verify the conclusions and to show the effectiveness of the algorithm.

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $\binom{n}{k}$ points in the $k$-slice (which consists of all $n$-bit strings with exactly $k$ ones).

We study the problem of efficiently computing on encoded data. More specifically, we study the question of low-bandwidth computation of functions $F:\mathbb{F}^k \to \mathbb{F}$ of some data $x \in \mathbb{F}^k$, given access to an encoding $c \in \mathbb{F}^n$ of $x$ under an error correcting code. In our model -- relevant in distributed storage, distributed computation and secret sharing -- each symbol of $c$ is held by a different party, and we aim to minimize the total amount of information downloaded from each party in order to compute $F(x)$. Special cases of this problem have arisen in several domains, and we believe that it is fruitful to study this problem in generality. Our main result is a low-bandwidth scheme to compute linear functions for Reed-Solomon codes, even in the presence of erasures. More precisely, let $\epsilon > 0$ and let $\mathcal{C}: \mathbb{F}^k \to \mathbb{F}^n$ be a full-length Reed-Solomon code of rate $1 - \epsilon$ over a field $\mathbb{F}$ with constant characteristic. For any $\gamma \in [0, \epsilon)$, our scheme can compute any linear function $F(x)$ given access to any $(1 - \gamma)$-fraction of the symbols of $\mathcal{C}(x)$, with download bandwidth $O(n/(\epsilon - \gamma))$ bits. In contrast, the naive scheme that involves reconstructing the data $x$ and then computing $F(x)$ uses $\Theta(n \log n)$ bits. Our scheme has applications in distributed storage, coded computation, and homomorphic secret sharing.

State estimation for legged locomotion over a dynamic rigid surface (DRS), which is a rigid surface moving in the world frame (e.g., ships, aircraft, and trains), remains an under-explored problem. This paper introduces an invariant extended Kalman filter that estimates the robot's pose and velocity during DRS locomotion by using common sensors of legged robots (e.g., inertial measurement units (IMU), joint encoders, and RDB-D camera). A key feature of the filter lies in that it explicitly addresses the nonstationary surface-foot contact point and the hybrid robot behaviors. Another key feature is that, in the absence of IMU biases, the filter satisfies the attractive group affine and invariant observation conditions, and is thus provably convergent for the deterministic continuous phases. The observability analysis is performed to reveal the effects of DRS movement on the state observability, and the convergence property of the hybrid, deterministic filter system is examined for the observable state variables. Experiments of a Digit humanoid robot walking on a pitching treadmill validate the effectiveness of the proposed filter under sensor noise and biases as well as large estimation errors and DRS movement.

In this paper we propose a method that estimates the $SE(3)$ continuous trajectories (orientation and translation) of the dynamic rigid objects present in a scene, from multiple RGB-D views. Specifically, we fit the object trajectories to cumulative B-Splines curves, which allow us to interpolate, at any intermediate time stamp, not only their poses but also their linear and angular velocities and accelerations. Additionally, we derive in this work the analytical $SE(3)$ Jacobians needed by the optimization, being applicable to any other approach that uses this type of curves. To the best of our knowledge this is the first work that proposes 6-DoF continuous-time object tracking, which we endorse with significant computational cost reduction thanks to our analytical derivations. We evaluate our proposal in synthetic data and in a public benchmark, showing competitive results in localization and significant improvements in velocity estimation in comparison to discrete-time approaches.