State-Space Models (SSMs) have proven to be powerful tools for online function approximation and for modeling long-range dependencies in sequential data. While recent methods such as HiPPO have demonstrated strong performance using a few polynomial bases, they remain limited by their reliance on closed-form solutions for specific, well-behaved bases. The SaFARi framework generalizes this approach, enabling the construction of SSMs from arbitrary frames, including non-orthogonal and redundant ones, thus allowing an infinite diversity of possible "species'' within the SSM family. In this paper, we introduce WaLRUS (Wavelets for Long-range Representation Using SSMs), a new species of SaFARi built from Daubechies wavelet frames. We instantiate two variants, scaled-Walrus and translated-Walrus, and show that their multiresolution and localized nature offers significant advantages in representing non-smooth and transient signals. We compare Walrus to HiPPO-based models and demonstrate improved accuracy, better numerical properties, and more efficient implementations for online function approximation tasks.

We study the interpolation capabilities of implicit neural representations (INRs) of images. In principle, INRs promise a number of advantages, such as continuous derivatives and arbitrary sampling, being freed from the restrictions of a raster grid. However, empirically, INRs have been observed to poorly interpolate between the pixels of the fit image; in other words, they do not inherently possess a suitable prior for natural images. In this paper, we propose to address and improve INRs’ interpolation capabilities by explicitly integrating image prior information into the INR architecture via deep decoder, a specific implementation of the deep image prior (DIP). Our method, which we call TITAN, leverages a residual connection from the input which enables integrating the principles of the grid-based DIP into the grid-free INR. Through super-resolution and computed tomography experiments, we demonstrate that our method significantly improves upon classic INRs, thanks to the induced natural image bias. We also find that by constraining the weights to be sparse, image quality and sharpness are enhanced, increasing the Lipschitz constant.

Seismic advances in generative AI algorithms for imagery, text, and other data types have led to the temptation to use AI-synthesized data to train next-generation models. Repeating this process creates an autophagous ("self-consuming") loop whose properties are poorly understood. We conduct a thorough analytical and empirical analysis using state-of-the-art generative image models of three families of autophagous loops that differ in how fixed or fresh real training data is available through the generations of training and whether the samples from previous-generation models have been biased to trade off data quality versus diversity. Our primary conclusion across all scenarios is that *without enough fresh real data in each generation of an autophagous loop, future generative models are doomed to have their quality (precision) or diversity (recall) progressively decrease.* We term this condition Model Autophagy Disorder (MAD), by analogy to mad cow disease, and show that appreciable MADness arises in just a few generations.

The first step toward investigating the effectiveness of a treatment via a randomized trial is to split the population into control and treatment groups then compare the average response of the treatment group receiving the treatment to the control group receiving the placebo. To ensure that the difference between the two groups is caused only by the treatment, it is crucial that the control and the treatment groups have similar statistics. Indeed, the validity and reliability of a trial are determined by the similarity of two groups' statistics. Covariate balancing methods increase the similarity between the distributions of the two groups' covariates. However, often in practice, there are not enough samples to accurately estimate the groups' covariate distributions. In this article, we empirically show that covariate balancing with the standardized means difference (SMD) covariate balancing measure, as well as Pocock and Simon's sequential treatment assignment method, are susceptible to worst case treatment assignments. Worst case treatment assignments are those admitted by the covariate balance measure, but result in highest possible ATE estimation errors. We developed an adversarial attack to find adversarial treatment assignment for any given trial. Then, we provide an index to measure how close the given trial is to the worst case. To this end, we provide an optimization-based algorithm, namely adversarial treatment assignment in treatment effect trials (ATASTREET), to find the adversarial treatment assignments.

Deep neural networks have become essential for numerous applications due to their strong empirical performance such as vision, RL, and classification. Unfortunately, these networks are quite difficult to interpret, and this limits their applicability in settings where interpretability is important for safety, such as medical imaging. One type of deep neural network is neural tangent kernel that is similar to a kernel machine that provides some aspect of interpretability. To further contribute interpretability with respect to classification and the layers, we develop a new network as a combination of multiple neural tangent kernels, one to model each layer of the deep neural network individually as opposed to past work which attempts to represent the entire network via a single neural tangent kernel. We demonstrate the interpretability of this model on two datasets, showing that the multiple kernels model elucidates the interplay between the layers and predictions.

High dimensionality poses many challenges to the use of data, from visualization and interpretation, to prediction and storage for historical preservation. Techniques abound to reduce the dimensionality of fixed-length sequences, yet these methods rarely generalize to variable-length sequences. To address this gap, we extend existing methods that rely on the use of kernels to variable-length sequences via use of the Recurrent Neural Tangent Kernel (RNTK). Since a deep neural network with ReLu activation is a Max-Affine Spline Operator (MASO), we dub our approach Max-Affine Spline Kernel (MASK). We demonstrate how MASK can be used to extend principal components analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) and apply these new algorithms to separate synthetic time series data sampled from second-order differential equations.