ABSTRACT This paper introduces V olterra Neural Ordinary Differential Equations (VNODE), a piecewise continuous V olterra Neural Network that integrates nonlinear V olterra filtering with continuous-time neural ordinary differential equations for image classification. Drawing inspiration from the visual cortex, where discrete event processing is interleaved with continuous integration, VNODE alternates between discrete V olterra feature extraction and ODE-driven state evolution. VNODE consistently outperforms state-of-the-art models with improved computational complexity as exemplified on benchmark datasets like CIFAR-10 and Imagenet-1K. Index T erms-- Neural ODEs, V olterra Neural Networks, Image Classification, Continuous-time Models 1. INTRODUCTION Over the past decade, deep learning has transformed signal and image processing. Driven by Convolutional Neural Networks, Transformers, and their variants, it has set benchmarks in image classification, action recognition, object detection, and many other computer vision tasks [1].

What types of differences among causal structures with latent variables are impossible to distinguish by statistical data obtained by probing each visible variable? If the probing scheme is simply passive observation, then it is well-known that many different causal structures can realize the same joint probability distributions. Even for the simplest case of two visible variables, for instance, one cannot distinguish between one variable being a causal parent of the other and the two variables sharing a latent common cause. However, it is possible to distinguish between these two causal structures if we have recourse to more powerful probing schemes, such as the possibility of intervening on one of the variables and observing the other. Herein, we address the question of which causal structures remain indistinguishable even given the most informative types of probing schemes on the visible variables. We find that two causal structures remain indistinguishable if and only if they are both associated with the same mDAG structure (as defined by Evans (2016)). We also consider the question of when one causal structure dominates another in the sense that it can realize all of the joint probability distributions that can be realized by the other using a given probing scheme. (Equivalence of causal structures is the special case of mutual dominance.) Finally, we investigate to what extent one can weaken the probing schemes implemented on the visible variables and still have the same discrimination power as a maximally informative probing scheme.

LLVM for Data Processing When Apache Spark launched Tungsten, there was a hint of incorporating LLVM into data processing pipeline to make use of modern CPU features and its superb machine code for performance boost. LLVM appears again with Weld, a new code generation project for data analytics. The project claims that Tensorflow, Spark, and Numpy can be accelerated up to 30x with just few operations of Weld! Interestingly, Matei Zaharia is in its contributor list. ETL Starter Kit Extract, Transform, Load (ETL) refers to a process in database usage and especially in data warehousing.

The Zero-suppressed Sentential Decision Diagram (ZSDD) is a recentlydiscovered tractable representation of Boolean functions. ZSDD subsumes theZero-suppressed Binary Decision Diagram (ZDD) as a strict subset, andsimilar to ZDD, it can perform several useful operations like model countingand Apply operations. We propose a top-down compilation algorithmfor ZSDD that represents sets of specific graph substructures, e.g.,matchings and simple paths of a graph. We experimentally confirm that theproposed algorithm is faster than other construction methods includingbottom-up methods and top-down methods for ZDDs, and the resulting ZSDDsare smaller than ZDDs representing the same graph substructures. We alsoshow that the size constructed ZSDDs can be bounded by the branch-width of thegraph. This bound is tighter than that of ZDDs.

The Sentential Decision Diagram (SDD) is a recently proposed representation of Boolean functions, containing Ordered Binary Decision Diagrams (OBDDs) as a distinguished subclass. While OBDDs are characterized by total variable orders, SDDs are characterized more generally by vtrees. As both OBDDs and SDDs have canonical representations, searching for OBDDs and SDDs of minimal size simplifies to searching for variable orders and vtrees, respectively. For OBDDs, there are effective heuristics for dynamic reordering, based on locally swapping variables. In this paper, we propose an analogous approach for SDDs which navigates the space of vtrees via two operations: one based on tree rotations and a second based on swapping children in a vtree. We propose a particular heuristic for dynamically searching the space of vtrees, showing that it can find SDDs that are an order-of-magnitude more succinct than OBDDs found by dynamic reordering.