Recovering a signal via a quadratic system of equations has gained intensive attention recently.

Then, combining Lemmas A.1 and A.2 with Eq. 1 and 2, we have null ˆ N f null We start with an intuitive explanation, and then turn to the formal proof. We show that w.h.p. we obtain We use the standard two's complement binary By Lemma A.4, there is = The weights in layers 2,...,k are all in { 1, 1} . We extend this result to real-valued functions. We now turn to the formal proof. Thus, this inequality is linear.

Gaussian process (GP) priors are non-parametric generative models with appealing modelling properties for Bayesian inference: they can model non-linear relationships through noisy observations, have closed-form expressions for training and inference, and are governed by interpretable hyperparameters. However, GP models rely on Gaussianity, an assumption that does not hold in several real-world scenarios, e.g., when observations are bounded or have extreme-value dependencies, a natural phenomenon in physics, finance and social sciences. Although beyond-Gaussian stochastic processes have caught the attention of the GP community, a principled definition and rigorous treatment is still lacking. In this regard, we propose a methodology to construct stochastic processes, which include GPs, warped GPs, Student-t processes and several others under a single unified approach. We also provide formulas and algorithms for training and inference of the proposed models in the regression problem. Our approach is inspired by layers-based models, where each proposed layer changes a specific property over the generated stochastic process. That, in turn, allows us to push-forward a standard Gaussian white noise prior towards other more expressive stochastic processes, for which marginals and copulas need not be Gaussian, while retaining the appealing properties of GPs. We validate the proposed model through experiments with real-world data.

In multi-agent domains (MADs), an agent's action may not just change the world and the agent's knowledge and beliefs about the world, but also may change other agents' knowledge and beliefs about the world and their knowledge and beliefs about other agents' knowledge and beliefs about the world. The goals of an agent in a multi-agent world may involve manipulating the knowledge and beliefs of other agents' and again, not just their knowledge/belief about the world, but also their knowledge about other agents' knowledge about the world. Our goal is to present an action language (mA+) that has the necessary features to address the above aspects in representing and RAC in MADs. mA+ allows the representation of and reasoning about different types of actions that an agent can perform in a domain where many other agents might be present -- such as world-altering actions, sensing actions, and announcement/communication actions. It also allows the specification of agents' dynamic awareness of action occurrences which has future implications on what agents' know about the world and other agents' knowledge about the world. mA+ considers three different types of awareness: full-, partial- awareness, and complete oblivion of an action occurrence and its effects. This keeps the language simple, yet powerful enough to address a large variety of knowledge manipulation scenarios in MADs. The semantics of mA+ relies on the notion of state, which is described by a pointed Kripke model and is used to encode the agent's knowledge and the real state of the world. It is defined by a transition function that maps pairs of actions and states into sets of states. We illustrate properties of the action theories, including properties that guarantee finiteness of the set of initial states and their practical implementability. Finally, we relate mA+ to other related formalisms that contribute to RAC in MADs.

We consider the problem of comparing probability densities between two groups. To model the complex pattern of the underlying densities, we formulate the problem as a nonparametric density hypothesis testing problem. The major difficulty is that conventional tests may fail to distinguish the alternative from the null hypothesis under the controlled type I error. In this paper, we model log-transformed densities in a tensor product reproducing kernel Hilbert space (RKHS) and propose a probabilistic decomposition of this space. Under such a decomposition, we quantify the difference of the densities between two groups by the component norm in the probabilistic decomposition. Based on the Bernstein width, a sharp minimax lower bound of the distinguishable rate is established for the nonparametric two-sample test. We then propose a penalized likelihood ratio (PLR) test possessing the Wilks' phenomenon with an asymptotically Chi-square distributed test statistic and achieving the established minimax testing rate. Simulations and real applications demonstrate that the proposed test outperforms the conventional approaches under various scenarios.