arXiv subjects

Akinori Tanaka

Publications and source records attributed to Akinori Tanaka.

At least 19 recordsLinked to original sources

arXiv2024

Understanding Diffusion Models by Feynman's Path Integral

Score-based diffusion models have proven effective in image generation and have gained widespread usage; however, the underlying factors contributing to the performance disparity between stochastic and deterministic (i.e., the probability flow ODEs) sampling schemes remain unclear. We introduce a novel formulation of diffusion models using Feynman's path integral, which is a formulation originally developed for quantum physics. We find this formulation providing comprehensive descriptions of score-based generative models, and demonstrate the derivation of backward stochastic differential equations and loss functions.The formulation accommodates an interpolating parameter connecting stochastic and deterministic sampling schemes, and we identify this parameter as a counterpart of Planck's constant in quantum physics. This analogy enables us to apply the Wentzel-Kramers-Brillouin (WKB) expansion, a well-established technique in quantum physics, for evaluating the negative log-likelihood to assess the performance disparity between stochastic and deterministic sampling schemes.

cs.LG↗

arXiv2023

Discriminator optimal transport

Within a broad class of generative adversarial networks, we show that discriminator optimization process increases a lower bound of the dual cost function for the Wasserstein distance between the target distribution $p$ and the generator distribution $p_G$. It implies that the trained discriminator can approximate optimal transport (OT) from $p_G$ to $p$.Based on some experiments and a bit of OT theory, we propose a discriminator optimal transport (DOT) scheme to improve generated images. We show that it improves inception score and FID calculated by un-conditional GAN trained by CIFAR-10, STL-10 and a public pre-trained model of conditional GAN by ImageNet.

stat.ML↗

arXiv2022

Bézier Flow: a Surface-wise Gradient Descent Method for Multi-objective Optimization

In this paper, we propose a strategy to construct a multi-objective optimization algorithm from a single-objective optimization algorithm by using the Bézier simplex model. Also, we extend the stability of optimization algorithms in the sense of Probability Approximately Correct (PAC) learning and define the PAC stability. We prove that it leads to an upper bound on the generalization with high probability. Furthermore, we show that multi-objective optimization algorithms derived from a gradient descent-based single-objective optimization algorithm are PAC stable. We conducted numerical experiments and demonstrated that our method achieved lower generalization errors than the existing multi-objective optimization algorithm.

math.OC↗

arXiv2021

Mott Insulator-like Bose-Einstein Condensation in a Tight-Binding System of Interacting Bosons with a Flat Band

We propose a new class of tight-binding systems of interacting bosons with a flat band, which are exactly solvable in the sense that one can explicitly write down the unique ground state. The ground state is expressed in terms of local creation operators, and apparently resembles that of a Mott insulator. Based on an exact representation in terms of a classical loop-gas model, we conjecture that the ground state may exhibit quasi Bose-Einstein condensation (BEC) or genuine BEC in dimensions two and three or higher, respectively, still keeping Mott insulator-like character. Our Monte Carlo simulation of the loop-gas model strongly supports this conjecture, i.e., the ground state undergoes a Kosterlitz-Thouless transition and exhibits quasi BEC in two dimensions.

cond-mat.quant-gas↗

arXiv2021

Approximate Bayesian Computation of Bézier Simplices

Bézier simplex fitting algorithms have been recently proposed to approximate the Pareto set/front of multi-objective continuous optimization problems. These new methods have shown to be successful at approximating various shapes of Pareto sets/fronts when sample points exactly lie on the Pareto set/front. However, if the sample points scatter away from the Pareto set/front, those methods often likely suffer from over-fitting. To overcome this issue, in this paper, we extend the Bézier simplex model to a probabilistic one and propose a new learning algorithm of it, which falls into the framework of approximate Bayesian computation (ABC) based on the Wasserstein distance. We also study the convergence property of the Wasserstein ABC algorithm. An extensive experimental evaluation on publicly available problem instances shows that the new algorithm converges on a finite sample. Moreover, it outperforms the deterministic fitting methods on noisy instances.

cs.LG↗

arXiv2020

Self-learning Monte-Carlo for non-abelian gauge theory with dynamical fermions

In this paper, we develop the self-learning Monte-Carlo (SLMC) algorithm for non-abelian gauge theory with dynamical fermions in four dimensions to resolve the autocorrelation problem in lattice QCD. We perform simulations with the dynamical staggered fermions and plaquette gauge action by both in HMC and SLMC for zero and finite temperature to examine the validity of SLMC. We confirm that SLMC can reduce autocorrelation time in non-abelian gauge theory and reproduces results from HMC. For finite temperature runs, we confirm that SLMC reproduces correct results with HMC, including higher-order moments of the Polyakov loop and the chiral condensate. Besides, our finite temperature calculations indicate that four flavor QC${}_2$D with $\hat{m} = 0.5$ is likely in the crossover regime in the Colombia plot.

hep-lat↗

arXiv2020

An extension of the cell-construction method for the flat-band ferromagnetism

We present an extension of the cell-construction method for the flat-band ferromagnetism. In a rather general setting, we construct Hubbard models with highly degenerate single-electron ground states and obtain a formal representation of these single-electron ground states. By our version of the cell-construction method, various types of flat-band Hubbard models, including the one on line graphs, can be designed and shown to have the unique ferromagnetic ground states when the electron number is equal to the degeneracy of the single-electron ground states.

cond-mat.str-el↗

arXiv2020

Self-learning Monte Carlo method with Behler-Parrinello neural networks

We propose a general way to construct an effective Hamiltonian in the Self-learning Monte Carlo method (SLMC), which speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. Its applications are, however, limited. This is because it is not obvious to find the explicit form of the effective Hamiltonians. Particularly, it is difficult to make trainable effective Hamiltonians including many-body interactions. In order to overcome this critical difficulty, we introduce the Behler-Parrinello neural networks (BPNNs) as "effective Hamiltonian'' without any prior knowledge, which is used to construct the potential-energy surfaces in interacting many particle systems for molecular dynamics. We combine SLMC with BPNN by focusing on a divisibility of Hamiltonian and propose how to construct the element-wise configurations. We apply it to quantum impurity models. We observed significant improvement of the acceptance ratio from 0.01 (the effective Hamiltonian with the explicit form) to 0.76 (BPNN). This drastic improvement implies that the BPNN effective Hamiltonian includes many body interaction, which is omitted in the effective Hamiltonian with the explicit forms. The BPNNs make SLMC more promising.

cond-mat.str-el↗

arXiv2019

Metric on random dynamical systems with vector-valued reproducing kernel Hilbert spaces

Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). We employ vvRKHSs to design mathematically manageable metrics and also to introduce operator-valued kernels, which enables us to handle randomness in systems. Our metric provides an extension of the existing metrics for deterministic systems, and gives a specification of the kernel maximal mean discrepancy of random processes. Moreover, by considering the time-wise independence of random processes, we clarify a connection between our metric and the independence criteria with kernels such as Hilbert-Schmidt independence criteria. We empirically illustrate our metric with synthetic data, and evaluate it in the context of the independence test for random processes. We also evaluate the performance with real time seris datas via clusering tasks.

stat.ML↗

arXiv2019

Asymptotic Risk of Bezier Simplex Fitting

The Bezier simplex fitting is a novel data modeling technique which exploits geometric structures of data to approximate the Pareto front of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from each skeleton of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a whole. In this paper, we analyze the asymptotic risks of those Bézier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a Bezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a Bezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a Bezier simplex of degree three where the all-at-once fitting gets an advantage.

cs.LG↗

arXiv2018

Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M-1)-dimensional topological simplex (a curved line for M=2, a curved triangle for M=3, a curved tetrahedron for M=4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bezier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bezier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the post-optimization process and enable a better trade-off analysis.

math.OC↗

arXiv2018

Deep Learning and Holographic QCD

We apply the relation between deep learning (DL) and the AdS/CFT correspondence to a holographic model of QCD. Using a lattice QCD data of the chiral condensate at a finite temperature as our training data, the deep learning procedure holographically determines an emergent bulk metric as neural network weights. The emergent bulk metric is found to have both a black hole horizon and a finite-height IR wall, so shares both the confining and deconfining phases, signaling the cross-over thermal phase transition of QCD. In fact, a quark antiquark potential holographically calculated by the emergent bulk metric turns out to possess both the linear confining part and the Debye screening part, as is often observed in lattice QCD. From this we argue the discrepancy between the chiral symmetry breaking and the quark confinement in the holographic QCD. The DL method is shown to provide a novel data-driven holographic modeling of QCD, and sheds light on the mechanism of emergence of the bulk geometries in the AdS/CFT correspondence.

hep-th↗

arXiv2018

Deep Learning and AdS/CFT

We present a deep neural network representation of the AdS/CFT correspondence, and demonstrate the emergence of the bulk metric function via the learning process for given data sets of response in boundary quantum field theories. The emergent radial direction of the bulk is identified with the depth of the layers, and the network itself is interpreted as a bulk geometry. Our network provides a data-driven holographic modeling of strongly coupled systems. With a scalar $ϕ^4$ theory with unknown mass and coupling, in unknown curved spacetime with a black hole horizon, we demonstrate our deep learning (DL) framework can determine them which fit given response data. First, we show that, from boundary data generated by the AdS Schwarzschild spacetime, our network can reproduce the metric. Second, we demonstrate that our network with experimental data as an input can determine the bulk metric, the mass and the quadratic coupling of the holographic model. As an example we use the experimental data of magnetic response of a strongly correlated material Sm$_{0.6}$Sr$_{0.4}$MnO$_3$. This AdS/DL correspondence not only enables gravity modeling of strongly correlated systems, but also sheds light on a hidden mechanism of the emerging space in both AdS and DL.

hep-th↗

arXiv2017

Towards reduction of autocorrelation in HMC by machine learning

In this paper we propose new algorithm to reduce autocorrelation in Markov chain Monte-Carlo algorithms for euclidean field theories on the lattice. Our proposing algorithm is the Hybrid Monte-Carlo algorithm (HMC) with restricted Boltzmann machine. We examine the validity of the algorithm by employing the phi-fourth theory in three dimension. We observe reduction of the autocorrelation both in symmetric and broken phase as well. Our proposing algorithm provides consistent central values of expectation values of the action density and one-point Green's function with ones from the original HMC in both the symmetric phase and broken phase within the statistical error. On the other hand, two-point Green's functions have slight difference between one calculated by the HMC and one by our proposing algorithm in the symmetric phase. Furthermore, near the criticality, the distribution of the one-point Green's function differs from the one from HMC. We discuss the origin of discrepancies and its improvement.

hep-lat↗

arXiv2017

Ferromagnetism in the Hubbard model with a gapless nearly-flat band

We present a version of the Hubbard model with a gapless nearly-flat lowest band which exhibits ferromagnetism in two or more dimensions. The model is defined on a lattice obtained by placing a site on each edge of the hypercubic lattice, and electron hopping is assumed to be only between nearest and next nearest neighbor sites. The lattice, where all the sites are identical, is simple, and the corresponding single-electron band structure, where two cosine-type bands touch without an energy gap, is also simple. We prove that the ground state of the model is unique and ferromagnetic at half-filling of the lower band, if the lower band is nearly flat and the strength of on-site repulsion is larger than a certain value which is independent of the lattice size. This is the first example of ferromagnetism in three dimensional non-singular models with a gapless band structure.

cond-mat.str-el↗

arXiv2017

Detection of phase transition via convolutional neural network

We design a Convolutional Neural Network (CNN) which studies correlation between discretized inverse temperature and spin configuration of 2D Ising model and show that it can find a feature of the phase transition without teaching any a priori information for it. We also define a new order parameter via the CNN and show that it provides well approximated critical inverse temperature. In addition, we compare the activation functions for convolution layer and find that the Rectified Linear Unit (ReLU) is important to detect the phase transition of 2D Ising model.

cond-mat.dis-nn↗

arXiv2016

Exact Path Integral for 3D Higher Spin Gravity

Extending the works arXiv:1504.05991 and arXiv:1510.02142, we study three dimensional Euclidean higher spin gravity with negative cosmological constant. This theory can be formulated in terms of SL(N,C) Chern-Simons theory. By introducing auxiliary fields, we rewrite it in a supersymmetric way and compute its partition function exactly by using the localization method. We obtain a good expression for the partition function in terms of characters for the vacuum and primaries in 2D unitary CFT with W_N symmetry. We also check that the coefficients of the character expansion are positive integers and exhibit Cardy formula in the large central charge limit.

hep-th↗

arXiv2016

One-dimensional extended Hubbard model with spin-triplet pairing ground states

We show that the one-dimensional extended Hubbard model has saturated ferromagnetic ground states with the spin-triplet electron pair condensation in a certain range of parameters. The ground state wave functions with fixed electron numbers are explicitly obtained. We also construct two ground states in which both the spin-rotation and the gauge symmetries are broken, and show that these states are transferred from one to the other by applying the edge operators. The edge operators are reduced to the Majorana fermions in a special case. These symmetry breaking ground states are shown to be stabilized by a superconducting mean field Hamiltonian which is related to the Kitaev chain with the charge-charge interaction.

cond-mat.str-el↗