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We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $b\Omega$ for some fixed $0 \leq p_0 \leq n$, $1 \leq q_0 \leq n$, or $M$ is a K\"ahler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_\epsilon$ that are transverse to the boundary $b\Omega$ and generate one forms, which when applied to $(p,q)$-forms, $0 \leq p \leq n$ and $q_0 \leq q \leq n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems., Comment: 28 pages. Comments welcome!

Morphogenesis is the process whereby the body of an organism develops its target shape. The morphogen BMP is known to play a conserved role across bilaterian organisms in determining the dorsoventral (DV) axis. Yet, how BMP governs the spatio-temporal dynamics of cytoskeletal proteins driving morphogenetic flow remains an open question. Here, we use machine learning to mine a morphodynamic atlas of Drosophila development, and construct a mathematical model capable of predicting the coupled dynamics of myosin, E-cadherin, and morphogenetic flow. Mutant analysis shows that BMP sets the initial condition of this dynamical system according to the following signaling cascade: BMP establishes DV pair-rule-gene patterns that set-up an E-cadherin gradient which in turn creates a myosin gradient in the opposite direction through mechanochemical feedbacks. Using neural tube organoids, we argue that BMP, and the signaling cascade it triggers, prime the conserved dynamics of neuroectoderm morphogenesis from fly to humans., Comment: Main text: 22 pages, 7 figures. Supplement: 18 pages, 10 figures

Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms., Comment: 29 pages. Comments welcome!

We evaluated the sensitivity of estimated PM2.5 and NO2 health impacts to varying key input parameters and assumptions including: 1) the spatial scale at which impacts are estimated, 2) using either a single concentration-response function (CRF) or using racial/ethnic group specific CRFs from the same epidemiologic study, 3) assigning exposure to residents based on home, instead of home and work locations. This analysis was carried out for the state of Colorado. We found that the spatial scale of the analysis influences the magnitude of NO2, but not PM2.5, attributable deaths. Using county-level predictions instead of 1 km2 predictions of NO2 resulted in a lower estimate of mortality attributable to NO2 by ~ 50% for all of Colorado for each year between 2000-2020. Using an all-population CRF instead of racial/ethnic group specific CRFs results in a higher estimate of annual mortality attributable to PM2.5 by a factor 1.3 for the white population and a lower estimate of mortality attributable to PM2.5 by factors of 0.4 and 0.8 for Black and Hispanic residents, respectively. Using racial/ethnic group specific CRFs did not result in a different estimation of NO2 attributable mortality for white residents, but led to lower estimates of mortality by a factor of ~ 0.5 for Black residents, and by a factor of 2.9 for to Hispanic residents. Using NO2 based on home instead of home and workplace locations results in a smaller estimate of annual mortality attributable to NO2 for all of Colorado by ~0.980 each year and 0.997 for PM2.5., Comment: 24 pages, 6 figures, 2 tables

The problem of foreground material signature extraction in an intimate (nonlinear) mixing setting is considered. It is possible for a foreground material signature to appear in combination with multiple background material signatures. We explore a framework for foreground material signature extraction based on a patch model that accounts for such background variation. We identify data conditions under which a foreground material signature can be extracted up to scaling and elementwise-inverse variations. We present algorithms based on volume minimization and endpoint member identification to recover foreground material signatures under these conditions. Numerical experiments on real and synthetic data illustrate the efficacy of the proposed algorithms., Comment: 41 pages, 5 figures

The Gemini North Adaptive Optics (GNAO) facility is the upcoming AO facility for Gemini North providing a state-of-the-art AO system for surveys and time domain science in the era of JWST and Rubin operations. GNAO will be optimized to feed the Gemini infrared Multi Object Spectrograph (GIRMOS). While GIRMOS is the primary science driver for defining the capabilities of GNAO, any instrument operating with an f/32 beam can be deployed using GNAO. The GNAO project includes the development of a new laser guide star facility which will consist of four side-launched laser beams supporting the two primary AO modes of GNAO: a wide-field mode providing an improved image quality over natural seeing for a 2-arcminute circular field-of-view and a narrow-field mode providing near diffraction-limited performance over a 20x20 arcsecond square field-of-view. The GNAO wide field mode will enable GIRMOS's multi-IFU configuration in which the science beam to each individual IFU will be additionally corrected using multi-object AO within GIRMOS. The GNAO narrow field mode will feed the GIRMOS tiled IFU configuration in which all IFUs are combined into a "super"-IFU in the center of the field. GNAO also includes the development of a new Real Time Controller, a new GNAO Facility System Controller and finally the development of a new AO Bench. We present in this paper an overview of the GNAO facility and provide a status update of each product., Comment: SPIE conference 2022 Montreal

The unit-modulus least squares (UMLS) problem has a wide spectrum of applications in signal processing, e.g., phase-only beamforming, phase retrieval, radar code design, and sensor network localization. Scalable first-order methods such as projected gradient descent (PGD) have recently been studied as a simple yet efficient approach to solving the UMLS problem. Existing results on the convergence of PGD for UMLS often focus on global convergence to stationary points. As a non-convex problem, only a sublinear convergence rate has been established. However, these results do not explain the fast convergence of PGD frequently observed in practice. This manuscript presents a novel analysis of convergence of PGD for UMLS, justifying the linear convergence behavior of the algorithm near the solution. By exploiting the local structure of the objective function and the constraint set, we establish an exact expression for the convergence rate and characterize the conditions for linear convergence. Simulations show that our theoretical analysis corroborates numerical examples. Furthermore, variants of PGD with adaptive step sizes are proposed based on the new insight revealed in our convergence analysis. The variants show substantial acceleration in practice., Comment: 16 pages

The second African Conference of Fundamental and Applied Physics (ACP2021) took place in the week of March 7-11, 2022. During this conference, all the African Strategy for Fundamental and Applied Physics (ASFAP) working groups had been reserved specials sessions to discuss their scope, activities (past & current) and topics of common interests. The aim of this report is to summarize the discussion of the ASFAP working groups in societal engagements, namely Physics Education, Community Engagement, Young Physicists and Women in Physics. The recommendations for future activities in societal engagements are summarised in the report as well., Comment: 7 pages, 2 figures Submitted to the Proceedings of the African Conference on Fundamental and Applied Physics Second Edition, ACP2021, March 7-11, 2022 - Virtual Event

It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case., Comment: 35 pages

Iterative hard thresholding (IHT) has gained in popularity over the past decades in large-scale optimization. However, convergence properties of this method have only been explored recently in non-convex settings. In matrix completion, existing works often focus on the guarantee of global convergence of IHT via standard assumptions such as incoherence property and uniform sampling. While such analysis provides a global upper bound on the linear convergence rate, it does not describe the actual performance of IHT in practice. In this paper, we provide a novel insight into the local convergence of a specific variant of IHT for matrix completion. We uncover the exact linear rate of IHT in a closed-form expression and identify the region of convergence in which the algorithm is guaranteed to converge. Furthermore, we utilize random matrix theory to study the linear rate of convergence of IHTSVD for large-scale matrix completion. We find that asymptotically, the rate can be expressed in closed form in terms of the relative rank and the sampling rate. Finally, we present various numerical results to verify the aforementioned theoretical analysis., Comment: 13 pages (plus Supplementary Material)

Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine learning and signal processing. This manuscript presents a unified framework for the local convergence analysis of projected gradient descent in the context of constrained least squares. The proposed analysis offers insights into pivotal local convergence properties such as the conditions for linear convergence, the region of convergence, the exact asymptotic rate of convergence, and the bound on the number of iterations needed to reach a certain level of accuracy. To demonstrate the applicability of the proposed approach, we present a recipe for the convergence analysis of projected gradient descent and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems, namely, linear equality-constrained least squares, sparse recovery, least squares with the unit norm constraint, and matrix completion., Comment: 21 pages including Supplementary Material

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations., Comment: 14 pages - final version accepted to OPTL22

In multiple instance multiple label learning, each sample, a bag, consists of multiple instances. To alleviate labeling complexity, each sample is associated with a set of bag-level labels leaving instances within the bag unlabeled. This setting is more convenient and natural for representing complicated objects, which have multiple semantic meanings. Compared to single instance labeling, this approach allows for labeling larger datasets at an equivalent labeling cost. However, for sufficiently large datasets, labeling all bags may become prohibitively costly. Active learning uses an iterative labeling and retraining approach aiming to provide reasonable classification performance using a small number of labeled samples. To our knowledge, only a few works in the area of active learning in the MIML setting are available. These approaches can provide practical solutions to reduce labeling cost but their efficacy remains unclear. In this paper, we propose a novel bag-class pair based approach for active learning in the MIML setting. Due to the partial availability of bag-level labels, we focus on the incomplete-label MIML setting for the proposed active learning approach. Our approach is based on a discriminative graphical model with efficient and exact inference. For the query process, we adapt active learning criteria to the novel bag-class pair selection strategy. Additionally, we introduce an online stochastic gradient descent algorithm to provide an efficient model update after each query. Numerical experiments on benchmark datasets illustrate the robustness of the proposed approach., Comment: Machine learning, Multiple instance multiple label learning, Active learning, incomplete label learning

Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis of the exact linear convergence rate of gradient descent for factorization-based matrix completion for symmetric matrices. Without any additional assumptions on the underlying model, we identify the deterministic condition for local convergence of gradient descent, which only depends on the solution matrix and the sampling set. More crucially, our analysis provides a closed-form expression of the asymptotic rate of convergence that matches exactly with the linear convergence observed in practice. To the best of our knowledge, our result is the first one that offers the exact rate of convergence of gradient descent for matrix factorization in Euclidean space for matrix completion., Comment: Full version

This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigate the geometric conditions on $M$ which the eigenvalue condition forces, 2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, 3) explore the $L^p$ and $L^p$-Sobolev mapping properties of the associated kernels, and 4) provide examples., Comment: 34 pages. Comments welcome!

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator $N$ and the projection onto the nullspace of $\Box_b$. The main application of our formulas is the critical case of codimension two quadrics in $\mathbb{C}^4$ where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci \cite{PeRi03}. We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups., Comment: 15 pages. Comments welcome!

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution., Comment: Accepted to Linear Algebra and Its Applications