Related papers: Complexity and Avoidance
A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over…
Constraints on the validity of the hierarchical gravitational instability theory and the evolution of biasing are presented based upon measurements of higher order clustering statistics in the Deeprange Survey, a catalog of $\sim710,000$…
We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an…
Using each node's degree as a proxy for its importance, the topological hierarchy of a complex network is introduced and quantified. We propose a simple dynamical process used to construct networks which are either maximally or minimally…
For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO…
We numerically analyze the complexity of unitary time-evolution and precursor operators in one- and two-qubit systems using the framework of Nielsen complexity geometry. We find that, as expected, the complexities of unitary time evolution…
Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which…
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum…
The error threshold transition in a stochastic (i.e. finite population) version of the quasispecies model of molecular evolution is studied using finite-size scaling. For the single-sharp-peak replication landscape, the deterministic model…
Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them…
Partial rigidity is a quantitative notion of recurrence and provides a global obstruction which prevents the system from being strongly mixing. A dynamical system $(X, \mathcal{X}, \mu, T)$ is partially rigid if there is a constant $\delta…
Traditional statistical analysis requires that the analysis process and data are independent. By contrast, the new field of adaptive data analysis hopes to understand and provide algorithms and accuracy guarantees for research as it is…
We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys…
The description complexity of a model is the length of the shortest formula that defines the model. We study the description complexity of unary structures in first-order logic FO, also drawing links to semantic complexity in the form of…
Since the time of Darwin, scientists have struggled to reconcile the evolution of biological forms in a universe determined by fixed laws. These laws underpin the origin of life, evolution, human culture and technology, as set by the…
Dynamic decision-making under distributional shifts is of fundamental interest in theory and applications of reinforcement learning: The distribution of the environment in which the data is collected can differ from that of the environment…
This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear…
We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
The stunning empirical successes of neural networks currently lack rigorous theoretical explanation. What form would such an explanation take, in the face of existing complexity-theoretic lower bounds? A first step might be to show that…