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Stefan Heinrich

Publications and source records attributed to Stefan Heinrich.

30 records · Page 2Linked to original sources

Interactive Natural Language Acquisition in a Multi-modal Recurrent Neural Architecture

For the complex human brain that enables us to communicate in natural language, we gathered good understandings of principles underlying language acquisition and processing, knowledge about socio-cultural conditions, and insights about activity patterns in the brain. However, we were not yet able to understand the behavioural and mechanistic characteristics for natural language and how mechanisms in the brain allow to acquire and process language. In bridging the insights from behavioural psychology and neuroscience, the goal of this paper is to contribute a computational understanding of appropriate characteristics that favour language acquisition. Accordingly, we provide concepts and refinements in cognitive modelling regarding principles and mechanisms in the brain and propose a neurocognitively plausible model for embodied language acquisition from real world interaction of a humanoid robot with its environment. In particular, the architecture consists of a continuous time recurrent neural network, where parts have different leakage characteristics and thus operate on multiple timescales for every modality and the association of the higher level nodes of all modalities into cell assemblies. The model is capable of learning language production grounded in both, temporal dynamic somatosensation and vision, and features hierarchical concept abstraction, concept decomposition, multi-modal integration, and self-organisation of latent representations.

cs.CL↗

On the randomized complexity of Banach space valued integration

We study the complexity of Banach space valued integration in the randomized setting. We are concerned with $r$-times continuously differentiable functions on the $d$-dimensional unit cube $Q$, with values in a Banach space $X$, and investigate the relation of the optimal convergence rate to the geometry of $X$. It turns out that the $n$-th minimal errors are bounded by $cn^{-r/d-1+1/p}$ if and only if $X$ is of equal norm type $p$.

math.NA↗

Quantum Lower Bounds by Entropy Numbers

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional L_p spaces and of Sobolev embeddings.

quant-ph↗

The Quantum Query Complexity of Elliptic PDE

The complexity of the following numerical problem is studied in the quantum model of computation: Consider a general elliptic partial differential equation of order 2m in a smooth, bounded domain Q\subset \R^d with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M\subseteq Q of dimension 0\le d_1 \le d. With the right hand side belonging to C^r(Q), and the error being measured in the L_\infty(M) norm, we prove that the n-th minimal quantum error is (up to logarithmic factors) of order n^{-min((r+2m)/d_1,r/d+1)}. For comparison, in the classical deterministic setting the n-th minimal error is known to be of order n^{-r/d}, for all d_1, while in the classical randomized setting it is (up to logarithmic factors) n^{-min((r+2m)/d_1,r/d+1/2)}.

quant-ph↗

On the Power of Quantum Algorithms for Vector Valued Mean Computation

We study computation of the mean of sequences with values in finite dimensional normed spaces and compare the computational power of classical randomized with that of quantum algorithms for this problem. It turns out that in contrast to the known superiority of quantum algorithms in the scalar case, in high dimensional $L_p^M$ spaces classical randomized algorithms are essentially as powerful as quantum algorithms.

quant-ph↗

Quantum Boolean Summation with Repetitions in the Worst-Average Setting

We study the quantum summation QS algorithm of Brassard, Hoyer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function defined on $N$ elements. We present sharp error bounds of the QS algorithm in the worst-average setting with the average performance measured in the $L_q$ norm, $q \in [1,\infty]$. We prove that the QS algorithm with $M$ quantum queries, $M<N$, has the worst-average error bounds of the form $Θ(\ln M/M)$ for $q=1$, $Θ(M^{-1/q})$ for $q\in (1,\infty)$, and is equal to 1 for $q=\infty$. We also discuss the asymptotic constants of these estimates. We improve the error bounds by using the QS algorithm with repetitions. Using the number of repetitions which is independent of $M$ and linearly dependent on $q$, we get the error bound of order $M^{-1}$ for any $q \in [1,\infty)$. Since $Ω(M^{-1})$ is a lower bound on the worst-average error of any quantum algorithm with $M$ queries, the QS algorithm with repetitions is optimal in the worst-average setting.

quant-ph↗

Quantum Approximation I. Embeddings of Finite Dimensional L_p Spaces

We study approximation of embeddings between finite dimensional L_p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper.

quant-ph↗

Quantum Approximation II. Sobolev Embeddings

A basic problem of approximation theory, the approximation of functions from the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out that in certain regions of the domain of parameters p,q,r,d quantum computation can reach a speedup of roughly squaring the rate of convergence of classical deterministic or randomized approximation methods. There are other regions were the best possible rates coincide for all three settings.

quant-ph↗

From Monte Carlo to Quantum Computation

Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this paper we give a short introduction to the basic ideas of quantum computing and survey recent results on high dimensional integration. We discuss connections to the Monte Carlo methology and compare the optimal error rates of quantum algorithms to those of classical deterministic and randomized algorithms.

quant-ph↗

Quantum Integration in Sobolev Classes

We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes $W^r_p([0,1]^d)$ and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from Hölder classes.

quant-ph↗

On a Problem in Quantum Summation

We consider the computation of the mean of sequences in the quantum model of computation. We determine the query complexity in the case of sequences which satisfy a $p$-summability condition for $1\le p<2$. This settles a problem left open in Heinrich (2001).

quant-ph↗

Quantum Summation with an Application to Integration

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a p-summability condition and for integration of functions from Lebesgue spaces L_p([0,1]^d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Brassard, Hoyer, Mosca, and Tapp (2000) on computing the mean for bounded sequences and complements results of Novak (2001) on integration of functions from Hoelder classes.

quant-ph↗