Spectral precision matrix, the inverse of a spectral density matrix, is an object of central interest in frequency-domain analysis of multivariate time series. Estimation of spectral precision matrix is a key step in calculating partial coherency and graphical model selection of stationary time series. When the dimension of a multivariate time series is moderate to large, traditional estimators of spectral density matrices such as averaged periodograms tend to be severely ill-conditioned, and one needs to resort to suitable regularization strategies involving optimization over complex variables. In this work, we propose complex graphical Lasso (CGLASSO), an $\ell_1$-penalized estimator of spectral precision matrix based on local Whittle likelihood maximization. We develop fast $\textit{pathwise coordinate descent}$ algorithms for implementing CGLASSO on large dimensional time series data sets. At its core, our algorithmic development relies on a ring isomorphism between complex and real matrices that helps map a number of optimization problems over complex variables to similar optimization problems over real variables. This finding may be of independent interest and more broadly applicable for high-dimensional statistical analysis with complex-valued data. We also present a complete non-asymptotic theory of our proposed estimator which shows that consistent estimation is possible in high-dimensional regime as long as the underlying spectral precision matrix is suitably sparse. We compare the performance of CGLASSO with competing alternatives on simulated data sets, and use it to construct partial coherence network among brain regions from a real fMRI data set.
相關內容
Face morphing is a problem in computer graphics with numerous artistic and forensic applications. It is challenging due to variations in pose, lighting, gender, and ethnicity. This task consists of a warping for feature alignment and a blending for a seamless transition between the warped images. We propose to leverage coord-based neural networks to represent such warpings and blendings of face images. During training, we exploit the smoothness and flexibility of such networks by combining energy functionals employed in classical approaches without discretizations. Additionally, our method is time-dependent, allowing a continuous warping/blending of the images. During morphing inference, we need both direct and inverse transformations of the time-dependent warping. The first (second) is responsible for warping the target (source) image into the source (target) image. Our neural warping stores those maps in a single network dismissing the need for inverting them. The results of our experiments indicate that our method is competitive with both classical and generative models under the lens of image quality and face-morphing detectors. Aesthetically, the resulting images present a seamless blending of diverse faces not yet usual in the literature.
The solution of a sparse system of linear equations is ubiquitous in scientific applications. Iterative methods, such as the Preconditioned Conjugate Gradient method (PCG), are normally chosen over direct methods due to memory and computational complexity constraints. However, the efficiency of these methods depends on the preconditioner utilized. The development of the preconditioner normally requires some insight into the sparse linear system and the desired trade-off of generating the preconditioner and the reduction in the number of iterations. Incomplete factorization methods tend to be black box methods to generate these preconditioners but may fail for a number of reasons. These reasons include numerical issues that require searching for adequate scaling, shifting, and fill-in while utilizing a difficult to parallelize algorithm. With a move towards heterogeneous computing, many sparse applications find GPUs that are optimized for dense tensor applications like training neural networks being underutilized. In this work, we demonstrate that a simple artificial neural network trained either at compile time or in parallel to the running application on a GPU can provide an incomplete sparse Cholesky factorization that can be used as a preconditioner. This generated preconditioner is as good or better in terms of reduction of iterations than the one found using multiple preconditioning techniques such as scaling and shifting. Moreover, the generated method also works and never fails to produce a preconditioner that does not reduce the iteration count.
Due to its high sample complexity, simulation is, as of today, critical for the successful application of reinforcement learning. Many real-world problems, however, exhibit overly complex dynamics, which makes their full-scale simulation computationally slow. In this paper, we show how to decompose large networked systems of many agents into multiple local components such that we can build separate simulators that run independently and in parallel. To monitor the influence that the different local components exert on one another, each of these simulators is equipped with a learned model that is periodically trained on real trajectories. Our empirical results reveal that distributing the simulation among different processes not only makes it possible to train large multi-agent systems in just a few hours but also helps mitigate the negative effects of simultaneous learning.
This paper intends to apply the sample-average-approximation (SAA) scheme to solve a system of stochastic equations (SSE), which has many applications in a variety of fields. The SAA is an effective paradigm to address risks and uncertainty in stochastic models from the perspective of Monte Carlo principle. Nonetheless, a numerical conflict arises from the sample size of SAA when one has to make a tradeoff between the accuracy of solutions and the computational cost. To alleviate this issue, we incorporate a gradually reinforced SAA scheme into a differentiable homotopy method and develop a gradually reinforced sample-average-approximation (GRSAA) differentiable homotopy method in this paper. By introducing a series of continuously differentiable functions of the homotopy parameter $t$ ranging between zero and one, we establish a differentiable homotopy system, which is able to gradually increase the sample size of SAA as $t$ descends from one to zero. The set of solutions to the homotopy system contains an everywhere smooth path, which starts from an arbitrary point and ends at a solution to the SAA with any desired accuracy. The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE. Several numerical experiments further confirm the effectiveness and efficiency of the proposed method.
Consider a committee election consisting of (i) a set of candidates who are divided into arbitrary groups each of size \emph{at most} two and a diversity constraint that stipulates the selection of \emph{at least} one candidate from each group and (ii) a set of voters who are divided into arbitrary populations each approving \emph{at most} two candidates and a representation constraint that stipulates the selection of \emph{at least} one candidate from each population who has a non-null set of approved candidates. The DiRe (Diverse + Representative) committee feasibility problem (a.k.a. the minimum vertex cover problem on unweighted undirected graphs) concerns the determination of the smallest size committee that satisfies the given constraints. Here, for this problem, we discover an unconditional deterministic polynomial-time algorithm that is an amalgamation of maximum matching, breadth-first search, maximal matching, and local minimization.
Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.
We take the classic facility location problem and consider a variation, in which each agent's individual cost function is equal to their distance from the facility multiplied by a scaling factor which is determined by the facility placement. In addition to the general class of continuous scaling functions, we also provide results for piecewise linear scaling functions which can effectively approximate or model the scaling of many real world scenarios. We focus on the objectives of total and maximum cost, describing the computation of the optimal solution. We then move to the approximate mechanism design setting, observing that the agents' preferences may no longer be single-peaked. Consequently, we characterize the conditions on scaling functions which ensure that agents have single-peaked preferences. Under these conditions, we find results on the total and maximum cost approximation ratios achievable by strategyproof and anonymous mechanisms.
Dense subgraph extraction is a fundamental problem in graph analysis and data mining, aimed at identifying cohesive and densely connected substructures within a given graph. It plays a crucial role in various domains, including social network analysis, biological network analysis, recommendation systems, and community detection. However, extracting a subgraph with the highest node similarity is a lack of exploration. To address this problem, we studied the Member Selection Problem and extended it with a dynamic constraint variant. By incorporating dynamic constraints, our algorithm can adapt to changing conditions or requirements, allowing for more flexible and personalized subgraph extraction. This approach enables the algorithm to provide tailored solutions that meet specific needs, even in scenarios where constraints may vary over time. We also provide the theoretical analysis to show that our algorithm is 1/3-approximation. Eventually, the experiments show that our algorithm is effective and efficient in tackling the member selection problem with dynamic constraints.
The problem of substructure characteristic modes is reformulated using a scattering matrix-based formulation, generalizing subregion characteristic mode decomposition to arbitrary computational tools. It is shown that the scattering formulation is identical to the classical formulation based on the background Green's function for lossless systems. The scattering formulation, however, opens a variety of new subregion scenarios unavailable within previous formulations, including cases with lumped or wave ports or subregions in circuits. Thanks to its scattering nature, the formulation is solver-agnostic with the possibility to utilize an arbitrary full-wave method.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191