This work presents several expected generalization error bounds based on the Wasserstein distance.

Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem of \emph{density estimation}, in which a learning algorithm is given i.i.d.

This work presents several expected generalization error bounds based on the Wasserstein distance.

Let p be an unknown and arbitrary probability distribution over [0, 1). We consider the problem of density estimation, in which a learning algorithm is given i.i.d.

Let p be an unknown and arbitrary probability distribution over [0,1) . We consider the problem of \emph{density estimation}, in which a learning algorithm is given i.i.d. The main contribution of this paper is a highly efficient density estimation algorithm for learning using a variable-width histogram, i.e., a hypothesis distribution with a piecewise constant probability density function. In more detail, for any k and \eps, we give an algorithm that makes \tilde{O}(k/\eps 2) draws from p, runs in \tilde{O}(k/\eps 2) time, and outputs a hypothesis distribution h that is piecewise constant with O(k \log 2(1/\eps)) pieces. With high probability the hypothesis h satisfies \dtv(p,h) \leq C \cdot \opt_k(p) \eps, where \dtv denotes the total variation distance (statistical distance), C is a universal constant, and \opt_k(p) is the smallest total variation distance between p and any k -piecewise constant distribution.

We introduce uncertainty-aware object instance segmentation (UncOS) and demonstrate its usefulness for embodied interactive segmentation. To deal with uncertainty in robot perception, we propose a method for generating a hypothesis distribution of object segmentation. We obtain a set of region-factored segmentation hypotheses together with confidence estimates by making multiple queries of large pre-trained models. This process can produce segmentation results that achieve state-of-the-art performance on unseen object segmentation problems. The output can also serve as input to a belief-driven process for selecting robot actions to perturb the scene to reduce ambiguity. We demonstrate the effectiveness of this method in real-robot experiments. Website: https://sites.google.com/view/embodied-uncertain-seg

Let p be an unknown and arbitrary probability distribution over [0, 1). We consider the problem of density estimation, in which a learning algorithm is given i.i.d.

Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem of \emph{density estimation}, in which a learning algorithm is given i.i.d. The main contribution of this paper is a highly efficient density estimation algorithm for learning using a variable-width histogram, i.e., a hypothesis distribution with a piecewise constant probability density function. In more detail, for any $k$ and $\eps$, we give an algorithm that makes $\tilde{O}(k/\eps 2)$ draws from $p$, runs in $\tilde{O}(k/\eps 2)$ time, and outputs a hypothesis distribution $h$ that is piecewise constant with $O(k \log 2(1/\eps))$ pieces. With high probability the hypothesis $h$ satisfies $\dtv(p,h) \leq C \cdot \opt_k(p) \eps$, where $\dtv$ denotes the total variation distance (statistical distance), $C$ is a universal constant, and $\opt_k(p)$ is the smallest total variation distance between $p$ and any $k$-piecewise constant distribution.

We propose and analyze a new vantage point for the learning of mixtures of Gaussians: namely, the PAC-style model of learning probability distributions introduced by Kearns et al. Here the task is to construct a hypothesis mixture of Gaussians that is statistically indistinguishable from the actual mixture generating the data; specifically, the KL-divergence should be at most epsilon. In this scenario, we give a poly(n/epsilon)-time algorithm that learns the class of mixtures of any constant number of axis-aligned Gaussians in n-dimensional Euclidean space. Our algorithm makes no assumptions about the separation between the means of the Gaussians, nor does it have any dependence on the minimum mixing weight. This is in contrast to learning results known in the ``clustering'' model, where such assumptions are unavoidable. Our algorithm relies on the method of moments, and a subalgorithm developed in previous work by the authors (FOCS 2005) for a discrete mixture-learning problem.