The P-GPFA latent space and the signal subspace of SNP-GPFA We show here that the signal subspace in the SNP-GPFA model looks nearly identical to standard P-GPFA run on trial-averaged data. This is shown for the same data analyzed in Figure 5 in the main text. The signal latent dimensionality is 5 for the SNP-GPFA model, and the latent dimensionality is 5 for the P-GFPA model. We show the first 3 PCs for clarity. This nearly identical pattern in the subspaces suggests that the SNP-GPFA model is an extension of the P-GPFA model on trial averaged data, providing the same signal subspace as well as additional information about the noise subspace (see Figure 5 in main text for more info).

Data-driven algorithm design is a promising, learning-based approach for beyond worst-case analysis of algorithms with tunable parameters. An important open problem is the design of computationally efficient data-driven algorithms for combinatorial algorithm families with multiple parameters. As one fixes the problem instance and varies the parameters, the "dual" loss function typically has a piecewise-decomposable structure, i.e. is well-behaved except at certain sharp transition boundaries. Motivated by prior empirical work, we initiate the study of techniques to develop efficient ERM learning algorithms for data-driven algorithm design by enumerating the pieces of the sum dual loss functions for a collection of problem instances. The running time of our approach scales with the actual number of pieces that appear as opposed to worst case upper bounds on the number of pieces. Our approach involves two novel ingredients - an output-sensitive algorithm for enumerating polytopes induced by a set of hyperplanes using tools from computational geometry, and an execution graph which compactly represents all the states the algorithm could attain for all possible parameter values. We illustrate our techniques by giving algorithms for pricing problems, linkage-based clustering and dynamic-programming based sequence alignment.

Analyzing the motion of multiple biological agents, be it cells or individual animals, is pivotal for the understanding of complex collective behaviors. With the advent of advanced microscopy, detailed images of complex tissue formations involving multiple cell types have become more accessible in recent years. However, deciphering the underlying rules that govern cell movements is far from trivial. Here, we present a novel deep learning framework for estimating the underlying equations of motion from observed trajectories, a pivotal step in decoding such complex dynamics. Our framework integrates graph neural networks with neural differential equations, enabling effective prediction of two-body interactions based on the states of the interacting entities. We demonstrate the efficacy of our approach through two numerical experiments. First, we used simulated data from a toy model to tune the hyperparameters. Based on the obtained hyperparameters, we then applied this approach to a more complex model with non-reciprocal forces that mimic the collective dynamics of the cells of slime molds. Our results show that the proposed method can accurately estimate the functional forms of two-body interactions - even when they are nonreciprocal - thereby precisely replicating both individual and collective behaviors within these systems.

We start by choosing the positional embedding E and construct a Transformer network that implements quantization of the input, contextual mapping of the quantized input, and value mapping of the context ids. 1. Choose the positional embedding E according to γ in Assumption 1.2.

It is based on a multiscale operator transformer, with time-conditioned layer norms that enable continuous-in-time evaluations. A novel training strategy leveraging the semi-group property of time-dependent PDEs to allow for significant scaling-up of the training data is also proposed.

Stochastic gradient descent (SGD) is perhaps the most prevalent optimization method in modern machine learning. Contrary to the empirical practice of sampling from the datasets without replacement and with (possible) reshuffling at each epoch, the theoretical counterpart of SGD usually relies on the assumption of sampling with replacement. It is only very recently that SGD using sampling without replacement - shuffled SGD - has been analyzed with matching upper and lower bounds. However, we observe that those bounds are too pessimistic to explain often superior empirical performance of data permutations (sampling without replacement) over vanilla counterparts (sampling with replacement) on machine learning problems. Through fine-grained analysis in the lens of primal-dual cyclic coordinate methods and the introduction of novel smoothness parameters, we present several results for shuffled SGD on smooth and non-smooth convex losses, where our novel analysis framework provides tighter convergence bounds over all popular shuffling schemes (IG, SO, and RR). Notably, our new bounds predict faster convergence than existing bounds in the literature - by up to a factor of O( n), mirroring benefits from tighter convergence bounds using component smoothness parameters in randomized coordinate methods. Lastly, we numerically demonstrate on common machine learning datasets that our bounds are indeed much tighter, thus offering a bridge between theory and practice.

Microsoft Edge Game Assist has worked its way through Microsoft's development cycle, and has been released for everybody. Even though we associate "Microsoft" with "Windows," Microsoft has numerous little platforms that it bolts features on to. Microsoft Edge Game Assist is one of these: It's a specialized hint tool for Game Bar, a Windows gaming feature that's been around for over half a decade with a steadily advancing feature set that includes performance tools, screen capture, and more. Instead of forcing you to stop what you're doing and start typing terms into search boxes, Game Assist "knows" what game you're playing and opens up what you might call a specialized hint browser. I went hands-on with Microsoft Edge Game Assist in January, where I launched it alongside Baldur's Gate 3 to see what sort of tips it could offer.

Adam is one of the most popular optimization algorithms in deep learning. However, it is known that Adam does not converge in theory unless choosing a hyperparameter, i.e., β

Motivated by the neuromorphic principles that regulate biological neural behaviors, PPLNs are ideal for processing data captured by event cameras, which are built to simulate neural activities in the human retina. We discuss how to represent the membrane potential of an artificial neuron by a parametric piecewise linear function with learnable coefficients. This design echoes the idea of building deep models from learnable parametric functions recently popularized by Kolmogorov-Arnold Networks (KANs). Experiments demonstrate the state-of-the-art performance of PPLNs in event-based and image-based vision applications, including steering prediction, human pose estimation, and motion deblurring.

We thank all reviewers for their careful reading of the paper, thoughtful feedback, and constructive suggestions. Each reviewer's major comments are addressed below. Reviewer 1. Thanks for your time and effort devoted to reviewing our submission, as well as for the positive comments Distinct novelties relative to Ref. [31] are: i) Algorithm: The present submission develops Following your suggestion, [31] will be discussed more thoroughly in the revised paper. We will respectfully disagree that it "makes more sense to take a decaying step-size." Due to space limitation, the focus of this paper was placed on analysis under both IID and Markovian data.