Related papers: Relativized depth

Motivated by results on generic-case complexity in group theory, we apply the ideas of effective Baire category and effective measure theory to study complexity classes of functions which are "fractionally computable" by a partial…

Vision and language tasks such as Visual Relation Detection and Visual Question Answering benefit from semantic features that afford proper grounding of language. The 3D depth of objects depicted in 2D images is one such feature. However it…

We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between…

In an attempt to better understand generalization in deep learning, we study several possible explanations. We show that implicit regularization induced by the optimization method is playing a key role in generalization and success of deep…

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…

Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge.…

Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable $k$-means and $k$-median clustering problems where the explanation is captured by a threshold…

Existing work on understanding deep learning often employs measures that compress all data-dependent information into a few numbers. In this work, we adopt a perspective based on the role of individual examples. We introduce a measure of…

A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…

The notion of statistical depth has been extensively studied in multivariate and functional data over the past few decades. In contrast, the depth on temporal point process is still under-explored. The problem is challenging because a point…

This paper investigates the impact of query topology on the difficulty of answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first contribution is to clarify the worst-case size of positive existential (PE),…

We study the role of depth in training randomly initialized overparameterized neural networks. We give a general result showing that depth improves trainability of neural networks by improving the conditioning of certain kernel matrices of…

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…

Kernel density estimation is a popular method for estimating unseen probability distributions. However, the convergence of these classical estimators to the true density slows down in high dimensions. Moreover, they do not define meaningful…

Ontology embeddings map classes, roles, and individuals in ontologies into $\mathbb{R}^n$, and within $\mathbb{R}^n$ similarity between entities can be computed or new axioms inferred. For ontologies in the Description Logic…

Convolutional rectifier networks, i.e. convolutional neural networks with rectified linear activation and max or average pooling, are the cornerstone of modern deep learning. However, despite their wide use and success, our theoretical…

Functional depth is the functional data analysis technique that orders a functional data set. Unlike the case of data on the real line, defining this order is non-trivial, and particularly, with functional data, there are a number of…

Along with the rapid development of deep learning in practice, the theoretical explanations for its success become urgent. Generalization and expressivity are two widely used measurements to quantify theoretical behaviors of deep learning.…

Universality results for equivariant neural networks remain rare. Those that do exist typically hold only in restrictive settings: either they rely on regular or higher-order tensor representations, leading to impractically high-dimensional…

We present a method for estimating the complexity of an image based on Bennett's concept of logical depth. Bennett identified logical depth as the appropriate measure of organized complexity, and hence as being better suited to the…