The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels. This paper defines and proves a set of joint covariance properties for spatio-temporal receptive fields in terms of spatio-temporal derivative operators applied to spatio-temporally smoothed image data under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations. Specifically, the derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. For this purpose, we also fundamentally extend the notion of scale-normalized derivatives to affine-normalized derivatives, that are computed based on spatial smoothing with affine Gaussian kernels, and analyze the covariance properties of the resulting affine-normalized derivatives for the affine group as well as for important subgroups thereof. We conclude with a geometric analysis, showing how the derived joint covariance properties make it possible to relate or match spatio-temporal receptive field responses, when observing, possibly moving, local surface patches from different views, under locally linearized perspective or projective transformations, as well as when observing different instances of spatio-temporal events, that may occur either faster or slower between different views of similar spatio-temporal events.
相關內容
A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are clear, they traditionally are used for describing only binary edge information, which limits their utility for more complex relational data. Decorated graphons were introduced to extend the graphon framework by incorporating richer relationships, such as edge weights and types. This specificity in modelling connections provides more granular insight into network dynamics. Yet, there are no existing inference techniques for decorated graphons. We develop such an estimation method, extending existing techniques from traditional graphon estimation to accommodate these richer interactions. We derive the rate of convergence for our method and show that it is consistent with traditional non-parametric theory when the decoration space is finite. Simulations confirm that these theoretical rates are achieved in practice. Our method, tested on synthetic and empirical data, effectively captures additional edge information, resulting in improved network models. This advancement extends the scope of graphon estimation to encompass more complex networks, such as multiplex networks and attributed graphs, thereby increasing our understanding of their underlying structures.
Generative diffusion models apply the concept of Langevin dynamics in physics to machine leaning, attracting a lot of interests from engineering, statistics and physics, but a complete picture about inherent mechanisms is still lacking. In this paper, we provide a transparent physics analysis of diffusion models, formulating the fluctuation theorem, entropy production, equilibrium measure, and Franz-Parisi potential to understand the dynamic process and intrinsic phase transitions. Our analysis is rooted in a path integral representation of both forward and backward dynamics, and in treating the reverse diffusion generative process as a statistical inference, where the time-dependent state variables serve as quenched disorder akin to that in spin glass theory. Our study thus links stochastic thermodynamics, statistical inference and geometry based analysis together to yield a coherent picture about how the generative diffusion models work.
A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction technique - yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.
We provide a novel dimension-free uniform concentration bound for the empirical risk function of constrained logistic regression. Our bound yields a milder sufficient condition for a uniform law of large numbers than conditions derived by the Rademacher complexity argument and McDiarmid's inequality. The derivation is based on the PAC-Bayes approach with second-order expansion and Rademacher-complexity-based bounds for the residual term of the expansion.
Qualitative properties and stability analysis of the mathematical model for a DC-DC electric circuit
This paper describes a simplified model of an electric circuit with a DC-DC converter and a PID-regulator as a system of integral differential equations with an identically singular matrix multiplying the higher derivative of the desired vector-function. We use theoretical results on integral and differential equations and their systems to prove solvability of such a model and analyze its stability.
A linear-time algorithm for generating auxiliary subgraphs for the 3-edge-connected components of a connected multigraph is presented. The algorithm uses an innovative graph contraction operation and makes only one pass over the graph. By contrast, the previously best-known algorithms make multiple passes over the graph to decompose it into its 2-edge-connected components or 2-vertex-connected components, then its 3-edge-connected components or 3-vertex-connected components, and then construct a cactus representation for the 2-cuts to generate the auxiliary subgraphs for the 3-edge-connected components.
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.
Dynamic data visualizations can convey large amounts of information over time, such as using motion to depict changes in data values for multiple entities. Such dynamic displays put a demand on our visual processing capacities, yet our perception of motion is limited. Several techniques have been shown to improve the processing of dynamic displays. Staging the animation to sequentially show steps in a transition and tracing object movement by displaying trajectory histories can improve processing by reducing the cognitive load. In this paper, We examine the effectiveness of staging and tracing in dynamic displays. We showed participants animated line charts depicting the movements of lines and asked them to identify the line with the highest mean and variance. We manipulated the animation to display the lines with or without staging, tracing and history, and compared the results to a static chart as a control. Results showed that tracing and staging are preferred by participants, and improve their performance in mean and variance tasks respectively. They also preferred display time 3 times shorter when staging is used. Also, encoding animation speed with mean and variance in congruent tasks is associated with higher accuracy. These findings help inform real-world best practices for building dynamic displays. The supplementary materials can be found at //osf.io/8c95v/
The optimization of yields in multi-reactor systems, which are advanced tools in heterogeneous catalysis research, presents a significant challenge due to hierarchical technical constraints. To this respect, this work introduces a novel approach called process-constrained batch Bayesian optimization via Thompson sampling (pc-BO-TS) and its generalized hierarchical extension (hpc-BO-TS). This method, tailored for the efficiency demands in multi-reactor systems, integrates experimental constraints and balances between exploration and exploitation in a sequential batch optimization strategy. It offers an improvement over other Bayesian optimization methods. The performance of pc-BO-TS and hpc-BO-TS is validated in synthetic cases as well as in a realistic scenario based on data obtained from high-throughput experiments done on a multi-reactor system available in the REALCAT platform. The proposed methods often outperform other sequential Bayesian optimizations and existing process-constrained batch Bayesian optimization methods. This work proposes a novel approach to optimize the yield of a reaction in a multi-reactor system, marking a significant step forward in digital catalysis and generally in optimization methods for chemical engineering.
We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191