Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system. Two distinct approaches, adapted from techniques in the literature, are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation. We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.
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In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic-elliptic system modelling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit-implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.
Undirected graphical models are widely used to model the conditional independence structure of vector-valued data. However, in many modern applications, for example those involving EEG and fMRI data, observations are more appropriately modeled as multivariate random functions rather than vectors. Functional graphical models have been proposed to model the conditional independence structure of such functional data. We propose a neighborhood selection approach to estimate the structure of Gaussian functional graphical models, where we first estimate the neighborhood of each node via a function-on-function regression and subsequently recover the entire graph structure by combining the estimated neighborhoods. Our approach only requires assumptions on the conditional distributions of random functions, and we estimate the conditional independence structure directly. We thus circumvent the need for a well-defined precision operator that may not exist when the functions are infinite dimensional. Additionally, the neighborhood selection approach is computationally efficient and can be easily parallelized. The statistical consistency of the proposed method in the high-dimensional setting is supported by both theory and experimental results. In addition, we study the effect of the choice of the function basis used for dimensionality reduction in an intermediate step. We give a heuristic criterion for choosing a function basis and motivate two practically useful choices, which we justify by both theory and experiments.
In accordance with the principle of "data minimization", many internet companies are opting to record less data. However, this is often at odds with A/B testing efficacy. For experiments with units with multiple observations, one popular data minimizing technique is to aggregate data for each unit. However, exact quantile estimation requires the full observation-level data. In this paper, we develop a method for approximate Quantile Treatment Effect (QTE) analysis using histogram aggregation. In addition, we can also achieve formal privacy guarantees using differential privacy.
Typical pipelines for model geometry generation in computational biomedicine stem from images, which are usually considered to be at rest, despite the object being in mechanical equilibrium under several forces. We refer to the stress-free geometry computation as the reference configuration problem, and in this work we extend such a formulation to the theory of fully nonlinear poroelastic media. The main steps are (i) writing the equations in terms of the reference porosity and (ii) defining a time dependent problem whose steady state solution is the reference porosity. This problem can be computationally challenging as it can require several hundreds of iterations to converge, so we propose the use of Anderson acceleration to speed up this procedure. Our evidence shows that this strategy can reduce the number of iterations up to 80\%. In addition, we note that a primal formulation of the nonlinear mass conservation equations is not consistent due to the presence of second order derivatives of the displacement, which we alleviate through adequate mixed formulations. All claims are validated through numerical simulations in both idealized and realistic scenarios.
Modern data mining applications require to perform incremental clustering over dynamic datasets by tracing temporal changes over the resulting clusters. In this paper, we propose A-Posteriori affinity Propagation (APP), an incremental extension of Affinity Propagation (AP) based on cluster consolidation and cluster stratification to achieve faithfulness and forgetfulness. APP enforces incremental clustering where i) new arriving objects are dynamically consolidated into previous clusters without the need to re-execute clustering over the entire dataset of objects, and ii) a faithful sequence of clustering results is produced and maintained over time, while allowing to forget obsolete clusters with decremental learning functionalities. Four popular labeled datasets are used to test the performance of APP with respect to benchmark clustering performances obtained by conventional AP and Incremental Affinity Propagation based on Nearest neighbor Assignment (IAPNA) algorithms. Experimental results show that APP achieves comparable clustering performance while enforcing scalability at the same time.
In today's data-centric world, where data fuels numerous application domains, with machine learning at the forefront, handling the enormous volume of data efficiently in terms of time and energy presents a formidable challenge. Conventional computing systems and accelerators are continually being pushed to their limits to stay competitive. In this context, computing near-memory (CNM) and computing-in-memory (CIM) have emerged as potentially game-changing paradigms. This survey introduces the basics of CNM and CIM architectures, including their underlying technologies and working principles. We focus particularly on CIM and CNM architectures that have either been prototyped or commercialized. While surveying the evolving CIM and CNM landscape in academia and industry, we discuss the potential benefits in terms of performance, energy, and cost, along with the challenges associated with these cutting-edge computing paradigms.
The system architecture controlling a group of robots is generally set before deployment and can be either centralized or decentralized. This dichotomy is highly constraining, because decentralized systems are typically fully self-organized and therefore difficult to design analytically, whereas centralized systems have single points of failure and limited scalability. To address this dichotomy, we present the Self-organizing Nervous System (SoNS), a novel robot swarm architecture based on self-organized hierarchy. The SoNS approach enables robots to autonomously establish, maintain, and reconfigure dynamic multi-level system architectures. For example, a robot swarm consisting of $n$ independent robots could transform into a single $n$-robot SoNS and then into several independent smaller SoNSs, where each SoNS uses a temporary and dynamic hierarchy. Leveraging the SoNS approach, we show that sensing, actuation, and decision-making can be coordinated in a locally centralized way, without sacrificing the benefits of scalability, flexibility, and fault tolerance, for which swarm robotics is usually studied. In several proof-of-concept robot missions -- including binary decision-making and search-and-rescue -- we demonstrate that the capabilities of the SoNS approach greatly advance the state of the art in swarm robotics. The missions are conducted with a real heterogeneous aerial-ground robot swarm, using a custom-developed quadrotor platform. We also demonstrate the scalability of the SoNS approach in swarms of up to 250 robots in a physics-based simulator, and demonstrate several types of system fault tolerance in simulation and reality.
We suggest employing graph sparsification as a pre-processing step for maxcut programs using the QUBO solver. Quantum(-inspired) algorithms are recognized for their potential efficiency in handling quadratic unconstrained binary optimization (QUBO). Given that maxcut is an NP-hard problem and can be readily expressed using QUBO, it stands out as an exemplary case to demonstrate the effectiveness of quantum(-inspired) QUBO approaches. Here, the non-zero count in the QUBO matrix corresponds to the graph's edge count. Given that many quantum(-inspired) solvers operate through cloud services, transmitting data for dense graphs can be costly. By introducing the graph sparsification method, we aim to mitigate these communication costs. Experimental results on classical, quantum-inspired, and quantum solvers indicate that this approach substantially reduces communication overheads and yields an objective value close to the optimal solution.
Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.
Although Transformer-based methods have significantly improved state-of-the-art results for long-term series forecasting, they are not only computationally expensive but more importantly, are unable to capture the global view of time series (e.g. overall trend). To address these problems, we propose to combine Transformer with the seasonal-trend decomposition method, in which the decomposition method captures the global profile of time series while Transformers capture more detailed structures. To further enhance the performance of Transformer for long-term prediction, we exploit the fact that most time series tend to have a sparse representation in well-known basis such as Fourier transform, and develop a frequency enhanced Transformer. Besides being more effective, the proposed method, termed as Frequency Enhanced Decomposed Transformer ({\bf FEDformer}), is more efficient than standard Transformer with a linear complexity to the sequence length. Our empirical studies with six benchmark datasets show that compared with state-of-the-art methods, FEDformer can reduce prediction error by $14.8\%$ and $22.6\%$ for multivariate and univariate time series, respectively. the code will be released soon.
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