Related papers: Generalized Krylov Complexity
Nonlinear acceleration algorithms improve the performance of iterative methods, such as gradient descent, using the information contained in past iterates. However, their efficiency is still not entirely understood even in the quadratic…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…
We apply a recently developed measure of multiscale complexity to the Gaussian model consisting of continuous spins with bilinear interactions for a variety of interaction matrix structures. We find two universal behaviors of the complexity…
Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We…
In this work, we find that the complexity of quantum many-body states, defined as a spread in the Krylov basis, may serve as a new probe that distinguishes topological phases of matter. We illustrate this analytically in one of the…
Networks are a powerful tool to model complex systems, and the definition of many Graph Neural Networks (GNN), Deep Learning algorithms that can handle networks, has opened a new way to approach many real-world problems that would be hardly…
In the present paper, we introduce and study oriented Getzler-Kapranov complexes. These complexes are generalizations of S. Merkulov's oriented graph complex. We investigate their relation to the cohomology of moduli spaces of complex and…
This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of…
We study the Parallel Replica Dynamics in a general setting. We introduce a trajectory fragment framework that can be used to design and prove consistency of Parallel Replica algorithms for generic Markov processes. We use our framework to…
We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity…
We argue that parameterized complexity is a useful tool with which to study global constraints. In particular, we show that many global constraints which are intractable to propagate completely have natural parameters which make them…
In this paper we develop flexible Krylov methods for efficiently computing regularized solutions to large-scale linear inverse problems with an $\ell_2$ fit-to-data term and an $\ell_p$ penalization term, for $p\geq 1$. First we approximate…
In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and…
Graph structure learning is a well-established problem that aims at optimizing graph structures adaptive to specific graph datasets to help message passing neural networks (i.e., GNNs) to yield effective and robust node embeddings. However,…
This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and…
We construct a generalized class of quantum gravity condensate states, that allows the description of continuum homogeneous quantum geometries within the full theory. They are based on similar ideas already applied to extract effective…
Multiple Kernel Learning is a conventional way to learn the kernel function in kernel-based methods. MKL algorithms enhance the performance of kernel methods. However, these methods have a lower complexity compared to deep learning models…
We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the…
The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion…
A geometric generalization of contraction theory called~$k$-contraction was recently developed using $k$-compound matrices. In this note, we focus on the relations between $k$-contraction and two other generalized contraction frameworks:…