We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d.

This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifiable in principle. Similar to no-free-lunch theorems for statistical inference, the present results clarify the inevitability of substantial assumptions for causal inference. An additional benefit of our topological approach is that it easily accommodates SCMs with infinitely many variables. We finally suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.

Efficiently sampling from un-normalized target distributions is a fundamental problem in scientific computing and machine learning. Traditional approaches such as Markov Chain Monte Carlo (MCMC) guarantee asymptotically unbiased samples from such distributions but suffer from computational inefficiency, particularly when dealing with high-dimensional targets, as they require numerous iterations to generate a batch of samples. In this paper, we introduce an efficient and scalable neural implicit sampler that overcomes these limitations. The implicit sampler can generate large batches of samples with low computational costs by leveraging a neural transformation that directly maps easily sampled latent vectors to target samples without the need for iterative procedures. To train the neural implicit samplers, we introduce two novel methods: the KL training method and the Fisher training method.

Let, y be a point in that intersection. Since, by definition, ˆr(x0,) is the radius of the smallest ball with 1/2 + probability mass of f(x0 + P) over all possible centers in Rk and ˆRis the radius of the smallest such ball centered at ˆf(x), we must have ˆr(x0,) ˆR. Consider the smallest ball B(z0,ˆr(x, 1)) that encloses at least 1/2 + 1 probability mass of f(x+ P). Since, r is the radius of the minimum enclosing ball that contains at least half of the points in Z, we have r ˆr(x, 1). Now, using the definition of ˆRand following the same reasoning as theorem 2, we can say that, d( ˆf(x), ˆf(x0)) βˆr(x0,) + ˆR (1 + β) ˆR.

The study of provable adversarial robustness has mostly been limited to classification tasks and models with one-dimensional real-valued outputs. We extend the scope of certifiable robustness to problems with more general and structured outputs like sets, images, language, etc.

Embed to control: A locally linear307 latent dynamics model for control from raw images.