The magnitude of a metric space is a recently-established invariant, providing a measure of the 'effective size' of a space across multiple scales while also capturing numerous geometrical properties. We develop a family of magnitude-based measures of the intrinsic diversity of latent representations, formalising a novel notion of dissimilarity between magnitude functions of finite metric spaces. Our measures are provably stable under perturbations of the data, can be efficiently calculated, and enable a rigorous multi-scale comparison of latent representations. We show the utility and superior performance of our measures in an experimental suite that comprises different domains and tasks, including the evaluation of diversity, the detection of mode collapse, and the evaluation of generative models for text, image, and graph data.
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We develop a practical way of addressing the Errors-In-Variables (EIV) problem in the Generalized Method of Moments (GMM) framework. We focus on the settings in which the variability of the EIV is a fraction of that of the mismeasured variables, which is typical for empirical applications. For any initial set of moment conditions our approach provides a "corrected" set of moment conditions that are robust to the EIV. We show that the GMM estimator based on these moments is root-n-consistent, with the standard tests and confidence intervals providing valid inference. This is true even when the EIV are so large that naive estimators (that ignore the EIV problem) are heavily biased with their confidence intervals having 0% coverage. Our approach involves no nonparametric estimation, which is especially important for applications with many covariates, and settings with multivariate or non-classical EIV. In particular, the approach makes it easy to use instrumental variables to address EIV in nonlinear models.
Optimal behaviours of a system to perform a specific task can be achieved by leveraging the coupling between trajectory optimization, stabilization, and design optimization. This approach is particularly advantageous for underactuated systems, which are systems that have fewer actuators than degrees of freedom and thus require for more elaborate control systems. This paper proposes a novel co-design algorithm, namely Robust Trajectory Control with Design optimization (RTC-D). An inner optimization layer (RTC) simultaneously performs direct transcription (DIRTRAN) to find a nominal trajectory while computing optimal hyperparameters for a stabilizing time-varying linear quadratic regulator (TVLQR). RTC-D augments RTC with a design optimization layer, maximizing the system's robustness through a time-varying Lyapunov-based region of attraction (ROA) analysis. This analysis provides a formal guarantee of stability for a set of off-nominal states. The proposed algorithm has been tested on two different underactuated systems: the torque-limited simple pendulum and the cart-pole. Extensive simulations of off-nominal initial conditions demonstrate improved robustness, while real-system experiments show increased insensitivity to torque disturbances.
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.
This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings are analysed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assigns preconditioners to matrices, in numerical linear algebra.Thus the known pre-conditioner maps are viewd as a quantum-channel in finite dimentions.
Solutions to differential equations, which are used to model physical systems, are computed numerically by solving a set of discretized equations. This set of discretized equations is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, $x_0$, and iterate the algorithm to obtain a sequence of solution vectors, $x_k$, which are approximations to the exact solution of the linear system, $x$. The iterative algorithm is said to converge to $x$, in the field of reals, if and only if $x_k$ converges to $x$ in the limit of $k \to \infty$. In this paper, we formally prove the asymptotic convergence of a particular class of iterative methods called the stationary iterative methods, in the Coq theorem prover. We formalize the necessary and sufficient conditions required for the iterative convergence, and extend this result to two classical iterative methods: the Gauss--Seidel method and the Jacobi method. For the Gauss--Seidel method, we also formalize a set of easily testable conditions for iterative convergence, called the Reich theorem, for a particular matrix structure, and apply this on a model problem of the one-dimensional heat equation. We also apply the main theorem of iterative convergence to prove convergence of the Jacobi method on the model problem.
Most diffusion models assume that the reverse process adheres to a Gaussian distribution. However, this approximation has not been rigorously validated, especially at singularities, where t=0 and t=1. Improperly dealing with such singularities leads to an average brightness issue in applications, and limits the generation of images with extreme brightness or darkness. We primarily focus on tackling singularities from both theoretical and practical perspectives. Initially, we establish the error bounds for the reverse process approximation, and showcase its Gaussian characteristics at singularity time steps. Based on this theoretical insight, we confirm the singularity at t=1 is conditionally removable while it at t=0 is an inherent property. Upon these significant conclusions, we propose a novel plug-and-play method SingDiffusion to address the initial singular time step sampling, which not only effectively resolves the average brightness issue for a wide range of diffusion models without extra training efforts, but also enhances their generation capability in achieving notable lower FID scores. Code and models are released at //github.com/PangzeCheung/SingDiffusion.
This work delves into the complexities of machine unlearning in the face of distributional shifts, particularly focusing on the challenges posed by non-uniform feature and label removal. With the advent of regulations like the GDPR emphasizing data privacy and the right to be forgotten, machine learning models face the daunting task of unlearning sensitive information without compromising their integrity or performance. Our research introduces a novel approach that leverages influence functions and principles of distributional independence to address these challenges. By proposing a comprehensive framework for machine unlearning, we aim to ensure privacy protection while maintaining model performance and adaptability across varying distributions. Our method not only facilitates efficient data removal but also dynamically adjusts the model to preserve its generalization capabilities. Through extensive experimentation, we demonstrate the efficacy of our approach in scenarios characterized by significant distributional shifts, making substantial contributions to the field of machine unlearning. This research paves the way for developing more resilient and adaptable unlearning techniques, ensuring models remain robust and accurate in the dynamic landscape of data privacy and machine learning.
This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the open-loop scenario, the interaction of heat and beam dynamics lacks exponential stability solely considering thermal effects. To confront this challenge, two types of boundary-type state feedback controllers are proposed: (i) employing static feedback controllers entirely and (ii) adopting a hybrid approach wherein the electrical controller dynamically enhances system dynamics. In both cases, solutions of the PDE systems demonstrate exponential stability through meticulously formulated Lyapunov functions with diverse multipliers. The proposed proof technique establishes a robust foundation for demonstrating the exponential stability of Finite-Difference-based model reductions as the discretization parameter approaches zero.
For the purpose of efficient and cost-effective large-scale data labeling, crowdsourcing is increasingly being utilized. To guarantee the quality of data labeling, multiple annotations need to be collected for each data sample, and truth inference algorithms have been developed to accurately infer the true labels. Despite previous studies having released public datasets to evaluate the efficacy of truth inference algorithms, these have typically focused on a single type of crowdsourcing task and neglected the temporal information associated with workers' annotation activities. These limitations significantly restrict the practical applicability of these algorithms, particularly in the context of long-term and online truth inference. In this paper, we introduce a substantial crowdsourcing annotation dataset collected from a real-world crowdsourcing platform. This dataset comprises approximately two thousand workers, one million tasks, and six million annotations. The data was gathered over a period of approximately six months from various types of tasks, and the timestamps of each annotation were preserved. We analyze the characteristics of the dataset from multiple perspectives and evaluate the effectiveness of several representative truth inference algorithms on this dataset. We anticipate that this dataset will stimulate future research on tracking workers' abilities over time in relation to different types of tasks, as well as enhancing online truth inference.
Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.
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