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We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for…

We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…

The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not…

We characterize the set of properties of Boolean-valued functions on a finite domain $\mathcal{X}$ that are testable with a constant number of samples. Specifically, we show that a property $\mathcal{P}$ is testable with a constant number…

We address the problem of understanding the variable abundance of 3-node and 4-node subgraphs (motifs) in complex networks from a dynamical point of view. As a criterion in the determination of the functional significance of a n-node…

We address the problem of improving the performance and in particular the sample complexity of deep neural networks by enforcing and guaranteeing invariances to symmetry transformations rather than learning them from data. Group-equivariant…

Many applications require the robustness, or ideally the invariance, of a neural network to certain transformations of input data. Most commonly, this requirement is addressed by either augmenting the training data, using adversarial…

The evolution of spin network states in loop quantum gravity can be described by introducing a time variable, defined by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution,…

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space…

Given an alphabet size $m\in\mathbb{N}$ thought of as a constant, and $\vec{k} = (k_1,\ldots,k_m)$ whose entries sum of up $n$, the $\vec{k}$-multi-slice is the set of vectors $x\in [m]^n$ in which each symbol $i\in [m]$ appears precisely…

We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that -- under minimal assumptions -- a closed set is invariant with respect to a stochastic control system if and only if it is…

The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a…

Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation…

Randomization tests rely on simple data transformations and possess an appealing robustness property. In addition to being finite-sample valid if the data distribution is invariant under the transformation, these tests can be asymptotically…

This paper addresses the issues of conservativeness and computational complexity of probabilistic robustness analysis. We solve both issues by defining a new sampling strategy and robustness measure. The new measure is shown to be much less…

The problem of detecting changes in covariance for a single pair of features has been studied in some detail, but may be limited in importance or general applicability. In contrast, testing equality of covariance matrices of a {\it set} of…

We study the recently introduced notion of output-input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. The subject of this paper is developing the theory of output-input…

We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site…

Memory consistency models have been developed to specify what values may be returned by a read given that, in a distributed system, memory operations may only be partially ordered. Before this work, consistency models were defined…

We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework…