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Hayato Saigo

Publications and source records attributed to Hayato Saigo.

At least 37 records · Page 2Linked to original sources

Local reservoir model for choice-based learning

Decision making based on behavioral and neural observations of living systems has been extensively studied in brain science, psychology, and other disciplines. Decision-making mechanisms have also been experimentally implemented in physical processes, such as single photons and chaotic lasers. The findings of these experiments suggest that there is a certain common basis in describing decision making, regardless of its physical realizations. In this study, we propose a local reservoir model to account for choice-based learning (CBL). CBL describes decision consistency as a phenomenon where making a certain decision increases the possibility of making that same decision again later, which has been intensively investigated in neuroscience, psychology, etc. Our proposed model is inspired by the viewpoint that a decision is affected by its local environment, which is referred to as a local reservoir. If the size of the local reservoir is large enough, consecutive decision making will not be affected by previous decisions, thus showing lower degrees of decision consistency in CBL. In contrast, if the size of the local reservoir decreases, a biased distribution occurs within it, which leads to higher degrees of decision consistency in CBL. In this study, an analytical approach on local reservoirs is presented, as well as several numerical demonstrations. Furthermore, a physical architecture for CBL based on single photons is discussed, and the effects of local reservoirs is numerically demonstrated. Decision consistency in human decision-making tasks and in recruiting empirical data are evaluated based on local reservoir. In summary, the proposed local reservoir model paves a path toward establishing a foundation for computational mechanisms and the systematic analysis of decision making on different levels.

stat.ML↗

Scalable photonic reinforcement learning by time-division multiplexing of laser chaos

Reinforcement learning involves decision making in dynamic and uncertain environments and constitutes a crucial element of artificial intelligence. In our previous work, we experimentally demonstrated that the ultrafast chaotic oscillatory dynamics of lasers can be used to solve the two-armed bandit problem efficiently, which requires decision making concerning a class of difficult trade-offs called the exploration-exploitation dilemma. However, only two selections were employed in that research; thus, the scalability of the laser-chaos-based reinforcement learning should be clarified. In this study, we demonstrated a scalable, pipelined principle of resolving the multi-armed bandit problem by introducing time-division multiplexing of chaotically oscillated ultrafast time-series. The experimental demonstrations in which bandit problems with up to 64 arms were successfully solved are presented in this report. Detailed analyses are also provided that include performance comparisons among laser chaos signals generated in different physical conditions, which coincide with the diffusivity inherent in the time series. This study paves the way for ultrafast reinforcement learning by taking advantage of the ultrahigh bandwidths of light wave and practical enabling technologies.

cs.ET↗

Space-homogeneous quantum walks on Z from the viewpoint of complex analysis

The subject of this paper is quantum walks, which are expected to simulate several kinds of quantum dynamical systems. In this paper, we define analyticity for quantum walks on Z. Almost all the quantum walks on $\mathbb{Z}$ which have been already studied are analytic. In the framework of analytic quantum walks, we can enlarge the theory of quantum walks. We obtain not only several generalizations of known results, but also new types of theorems. It is proved that every analytic space-homogeneous quantum walk on Z is essentially a composite of shift operators and continuous-time analytic space-homogeneous quantum walks. We also prove existence of the weak limit distribution for analytic space-homogeneous quantum walks on Z.

math-ph↗

Meanings, Metaphors, and Morphisms: Theory of Indeterminate Natural Transformation (TINT)

In the present paper, we propose a new theory named "Theory of indeterminate natural transformation (TINT)" to investigate the dynamical creation of meanings as association relationships between images, focusing on the metaphor comprehension as an example. TINT models the meaning creation as a kind of stochastic processes based on the mathematical structure defined by association relationships as morphisms in category theory, so as to represent the indeterminate nature of structure-structure interactions between the systems of the meanings of images. Such interactions are formulated in terms of so-called coslice categories and functors as structure-preserving correspondence between them. The relationship between such functors is "indeterminate natural transformation", the central notion in TINT, which models the creation of meanings in a precise manner. For instance, the process of metaphor comprehension is modeled by the construction of indeterminate natural transformation from a canonically defined functor which we call the base-of-metaphor functor.

math.CT↗

On a property of the simple random walk on $\mathbb{Z}$

The subject of this paper is the simple random walk on $\mathbb{Z}$. We give a very simple answer to the following problem: under the condition that a random walk has already spent $α$-percent of the traveling time on the positive side $\mathbb{Z}_{\ge 0}$, what is the probability that the random walk is now on the positive side? The symmetric random walks which step $2n$-times can be decomposed in the following two ways: (1) how many times the walk steps on the positive side, (2) whether the last step is on the positive side or on the negative side. To answer the problem above, we clarify the number of the walks classified by (1) and (2). It has been already known that the distribution of the number indicated by (1) makes the arcsine law. Combining with the decomposition with respect to (2), we obtain a decomposition of the arcsine law into the Marchenko-Pastur law.

math.PR↗

Local state and sector theory in local quantum physics

We define a new concept of local states in the framework of algebraic quantum field theory (AQFT). Local states are a natural generalization of states and give a clear vision of localization in the context of QFT. In terms of them, we can find a condition from which follows automatically the famous DHR selection criterion in DHR-DR theory. As a result, we can understand the condition as consequences of physically natural state preparations in vacuum backgrounds. Furthermore, a theory of orthogonal decomposition of completely positive (CP) maps is developed. It unifies a theory of orthogonal decomposition of states and order structure theory of CP maps. By using it, localized version of sectors is formulated, which gives sector theory for local states with respect to general reference representations.

math-ph↗

Photon localization revisited

In the light of Newton-Wigner-Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as Quantum Field Ontology associated with Eventualization Principle, which enables us to explain the mutual relations back and forth, between quantum fields and various forms of particles in the localized form of the former.

quant-ph↗

The Arcsine law and an asymptotic behavior of orthogonal polynomials

Interacting Fock space connects the study of quantum probability theory, classical random variables, and orthogonal polynomials. It is a pre-Hilbert space associated with creation, preservation, and annihilation processes. We prove that if three processes are asymptotically commutative, the arcsine law arises as the "large quantum number limits." As a corollary, it is shown that for many probability measures, asymptotic behavior of orthogonal polynomials is described by the arcsine function. A weaker form of asymptotic commutativity provides us a discretized arcsine law.

math-ph↗

On operator-valued monotone independence

We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of Central Limit Theorem for operator-valued case. Moreover, we prove a generalization of Muraki's formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.

math.OA↗

Arcsine Law as the Classical Limit for interacting Fock spaces

In the present paper we discuss how to generalize ``Quantum-Classical Correspondence'' by means of the notion of interacting Fock spaces, which associates algebraic probability theory and the theory of orthogonal polynomials of probability measures. As an application we show that the Arcsine Law is ``Classical Limit'' for interacting Fock spaces corresponnding to certain kind of symmetric probability measures such as q-Gaussians. We also discuss the case of the exponential distribution as a simple example of asymmetric probability measures.

math.PR↗

Indeterminacy of the moment problem for symmetric probability measures

In this paper, the moment problem for symmetric probability measures is characterized in terms of associated sequences called Jacobi sequences $\{ω_n\}$. A notion named property (SC), which is proved to be a necessary and sufficient condition for the indeterminacy of the moment problem, naturally arises from the viewpoint of finite dimensional approximation for infinite matrices. We prove that the moment problem for q-Gaussian not only for $q>1$ but also for $q<-1$ is indeterminate. We also prove that hyperbolic secant distribution is "the last probability measure" which is uniquely determined by the moment sequence of power type, just by checking property (SC) with quite easy calculation.

math.FA↗

Notes on the Krupa-Zawisza Ultrapower of Self-Adjoint Operators

It is known that there is a difficulty in constructing the ultrapower of unbounded operators. Krupa and Zawisza gave a rigorous definition of the ultrapower A^{omega} of a selfadjoint operator A. In this note, we give alternative description of A^{omega} and the Hilbert space H(A) on which A^{omega} is densely defined, which provides a criterion to determine to which representing sequence (ξ_n)n of a given vector ξin dom(A^{omega}) has the property that A^{omega}ξ= (Aξ_n)_{omega} holds.

math.FA↗

Derivation of Born Rule from Algebraic and Statistical Axioms

In the present paper we propose a new axiomatic system of algebraic and statistical axioms as working hypotheses, from which Born rule can be seen to emerge. In this process the concept of sectors defined as quasi-equivalence classes of factor states plays a crucial role.

quant-ph↗

No Zero Divisor for Wick Product in $(S)^{\ast}$}

In White Noise Analysis (WNA), various random quantities are analyzed as elements of $(S)^{\ast}$, the space of Hida distributions ([1]). Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On $(S)^{\ast}$, the Wick product is defined in terms of the $\mathcal{S}$-transform. We have found such a remarkable property that the Wick product has no zero devisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the \textquotedblleft operational calculus\textquotedblright in WNA along the line of Mikusiński's version for solving differential equations.

math-ph↗

A New Look at the Arcsine Law and "Quantum-Classical Correspondence"

We prove that the Arcsine law as the time-averaged distribution for classical harmonic oscillators emerges from the distributions for quantum harmonic oscillators in terms of noncommutative algebraic probability. This is nothing but a simple and rigorous realization of "Quantum-Classical Correspondence" for harmonic oscillators.

math-ph↗

Joint cumulants for natural independence

Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigate relations between generating functions of moments and monotone cumulants. We find a natural extension of the Muraki formula, which describes the sum of monotone independent random variables, to the multivariate case.

math.OA↗

Who has seen a free photon?

While the notion of the position of photons is indispensable in the quantum optical situations, it has been known in mathematical physics that any position operator cannot be defined for a massless free particle with a non-zero finite spin. This dilemma is resolved by introducing the "effective mass" of a photon due to the interaction with matter. The validity of this interpretation is confirmed in reference to the picture of "polariton", a basic notion in optical and solid physics. In this connection, we discuss the long-standing controversy between Minkowski's and Abraham's definitions of the momenta of a photon in media from the general viewpoint adopted in the appendix.

physics.gen-ph↗