High-dimensional Space Research Articles

Transformation of valence signaling in a mouse striatopallidal circuit.

The ways in which sensory stimuli acquire motivational valence through association with other stimuli is one of the simplest forms of learning. Although we have identified many brain nuclei that play various roles in reward processing, a significant gap remains in understanding how valence encoding transforms through the layers of sensory processing. To address this gap, we carried out a comparative investigation of the mouse anteromedial olfactory tubercle (OT), and the ventral pallidum (VP) - 2 connected nuclei of the basal ganglia which have both been implicated in reward processing. First, using anterograde and retrograde tracing, we show that both D1 and D2 neurons of the anteromedial OT project primarily to the VP and minimally elsewhere. Using two-photon calcium imaging, we then investigated how the identity of the odor and reward contingency of the odor are differently encoded by neurons in either structure during a classical conditioning paradigm. We find that VP neurons robustly encode reward contingency, but not identity, in low-dimensional space. In contrast, the OT neurons primarily encode odor identity in high-dimensional space. Although D1 OT neurons showed larger responses to rewarded odors than other odors, consistent with prior findings, we interpret this as identity encoding with enhanced contrast. Finally, using a novel conditioning paradigm that decouples reward contingency and licking vigor, we show that both features are encoded by non-overlapping VP neurons. These results provide a novel framework for the striatopallidal circuit in which a high-dimensional encoding of stimulus identity is collapsed onto a low-dimensional encoding of motivational valence.

Deep reinforcement learning-based multi-objective optimization for electricity–gas–heat integrated energy systems

With the increasing global attention on energy efficiency and carbon emissions, the optimization of integrated energy systems (IES) has become the key to improve energy efficiency and reduce pollution emissions. However, most of the existing optimization methods cannot effectively deal with the complexity of high dimensional continuous action space. Therefore, this paper focuses on a novel multi-objective optimization strategy for the electricity–gas–heat integrated energy systems (EGH-IES). Firstly, considering the absorption capacity of wind power and the emission of pollutant gases, a multi-objective optimization model is constructed based on the mechanism model and operation constraints of each device in EGH-IES, in which the integrated operation cost and the environmental factors are taken as optimization objectives. Then, the multi-objective optimization problem is designed as the optimal strategy of interaction learning between agent and environment in reinforcement learning, and the output power of the devices constitutes the action of reinforcement learning. Additionally, the Ornstein–Uhlenbeck process is introduced to enhance the training efficiency and exploration performance of the agent, and the deep deterministic policy gradients (DDPG) algorithm is employed to optimize the action, thus the output power of the appliances could be obtained. Finally, the simulation results show that compared with deep Q network (DQN) method and proximal policy optimization (PPO) method, the reward function value of the proposed method increases by 2.43% and 6.09%, respectively, which represents a reduction in economic cost and pollutant emissions. These verify the effectiveness and superiority of the proposed multi-objective optimization scheme in cost reduction and benefit improvement for the EGH-IES.

Improving information theory of context-aware phrase embeddings in HR domain

The object of the study is context-aware phrase representations. The growing need to automate candidate recruitment and job recommendation processes has paved the way for the utilization of text embeddings. These embeddings involve translating the semantic essence of text into a continuous, high-dimensional vector space. By learning context-aware rich and meaningful representations of phrases within the human resource domain, the efficacy of similarity searches and matching procedures is enhanced, which contributes to a more streamlined and effective recruitment process. However, existing approaches do not take into account the context when modeling phrases. This necessitates the improvement of information technology analysis in this area. In this paper, it is proposed to mark the beginning and end of phrases in the text using special tokens. This made it possible to reduce the requirements for computing power by calculating all phrase representations present in the text simultaneously. The effectiveness of the improvement was tested on a new dataset to compare and evaluate the models in the task of modeling phrases in the field of human resources management. The proposed approach to modeling phrase representations with regard to context in the field of human resources management leads to an improvement in computational efficiency by up to 50 % and an increase in accuracy by up to 10 %. The architecture of the machine learning model for creating context-aware phrase representations is developed, which is characterized by the presence of blocks for taking into account phrase boundaries. Experiments and comparisons with existing approaches have confirmed the effectiveness of the proposed solution. In practice, the proposed information analysis technology can be used to automate the process of identifying and normalizing candidates' skills in online recruiting

Computation and Applications of Gaussian Integrals in Mathematics and Applied Sciences

Gaussian integrals, particularly those involving the function e−x2 , play a central role in various fields, ranging from physics to finance. This paper explores the computation of Gaussian integrals, beginning with the fundamental integral in the interval −∞ to ∞, which yields π . The author extends this analysis to integrals involving powers of x multiplied by e−x2 , showing their relevance in calculating moments in quantum mechanics and probability theory. Additionally, the author discusses the applications of multivariate Gaussian integrals in machine learning and statistical mechanics, where they are key to solving problems in high-dimensional spaces. Practical examples are provided from quantum mechanics (path integrals), statistical mechanics (partition functions), finance (option pricing models), and machine learning (Gaussian processes). Through these examples, the paper highlights the versatility and universality of Gaussian integrals as essential tools in both theoretical and applied contexts. The integral’s widespread applicability reflects its importance in connecting mathematical theory with real-world phenomena. This work highlights the role played by the Gaussian integral.

Jet mixing optimization using a bio-inspired evolution of hardware and control

Jet mixing is a critical factor in various engineering applications, influencing pollutant dispersion, chemical processes, medical treatments, and combustion enhancement. Hitherto, jet mixing has typically been optimized by either passive or active control techniques. In this experimental study, we combine simultaneous optimization of active control with 12 inward-pointing minijets and a tuneable nozzle exit shape commanded by 12 stepper motors. Jet mixing is monitored at the end of the potential core with an array of 7 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ imes$$\\end{document} 7 Pitot tubes. This high-dimensional actuation space is conquered with Particle Swarm Optimization through Targeted, Position-Mutated Elitism. Our results underscore the significant impact of combining control techniques, illustrating the complex interactions of both passive and active control on jet flow dynamics. The mixing area of the combined control optimization is 4.5 times larger than the area of the unforced state. This mixing increase significantly outperforms the effect of shape optimization of the nozzle alone. Our study points at the potential of optimization in high-dimensional design spaces for shapes as well as passive and active control—leveraging the rapid development of flow control hardware and the increasingly powerful tools of artificial intelligence for optimization.

A neural network approach for unstructured mesh quality evaluation

PurposeThe quality of the unstructured mesh has a considerable impact on the stability and accuracy of aerodynamic simulation in computational fluid dynamics (CFD). Typically, engineers spend a significant portion of their time on mesh quality evaluation to ensure a valid, high-quality mesh. The extensive manual interaction and a priori knowledge required to undertake an accurate and timely evaluation process have become a bottleneck in the idealized efficient CFD workflow. This paper aims to introduce a neural network-based quality evaluation approach for unstructured meshes to enable higher efficiency and the level of automation.Design/methodology/approachThe paper investigates the capability of deep neural networks for the quality evaluation of unstructured meshes. For training the network, we build a training dataset for mesh quality learning algorithms. The dataset contains a rich variety of unstructured aircraft meshes with different mesh sizes, densities, cell distribution, growth ratios and cell numbers to ensure its diversity and availability. We also design a neural network, AircraftNet, to learn the effect of mesh quality on the convergent properties of the numerical solutions. The proposed network directly manipulates raw point data in mesh source files rather than passing it to an intermediate data representation. During training, AircraftNet extracts non-linear quality features from high-dimensional data spaces and then automatically predicts the overall quality of the input unstructured mesh.FindingsThe paper provides a series of experimental results on GPUs. It shows that AircraftNet is able to effectively analyze the quality-related features like mesh density and distribution from the extracted features and achieve high prediction accuracy on the proposed dataset with even a small number of training runs.Research limitations/implicationsBecause of the limited training dataset, the research results may lack generalizability. Therefore, researchers are encouraged to test the proposed propositions further.Originality/valueThe paper publishes a benchmarking dataset for mesh quality learning algorithms and designs a novel neural network approach for unstructured mesh quality evaluation.

Variational Bayesian Monte Carlo simulation for rare event assessment

Estimating rare events in structural engineering demands intensive computational resources, and the complexity of the performance function poses challenges to accurate probability estimation. To efficiently conduct the structural reliability analysis on rare events, we propose a novel variational Bayesian Monte Carlo (VBMC)-based subset simulation (SS) method. The proposed approach can be divided into two aspects: the Monte Carlo simulation (MCS) is first applied to estimate the first-layer unconditional failure probability, then VBMC is employed to estimate subsequent conditional failure probabilities. Finally, the original small failure probability is obtained by the product of the first-layer unconditional failure probability and a series of conditional failure probabilities. The proposed approach inherits the merits of SS as well as VBMC, which converts the tricky rare event estimation into a series of high-frequency event estimations and leverages VBMC instead of the canonical Markov chain to produce the independent and informative next-generation failure-conditional samples. Four case studies, including high-dimensional and discrete state space scenarios, are performed to illustrate the feasibility and generality of the proposed method.

Mcadet: A feature selection method for fine-resolution single-cell RNA-seq data based on multiple correspondence analysis and community detection.

Single-cell RNA sequencing (scRNA-seq) data analysis faces numerous challenges, including high sparsity, a high-dimensional feature space, and biological noise. These challenges hinder downstream analysis, necessitating the use of feature selection methods to identify informative genes, and reduce data dimensionality. However, existing methods for selecting highly variable genes (HVGs) exhibit limited overlap and inconsistent clustering performance across benchmark datasets. Moreover, these methods often struggle to accurately select HVGs from fine-resolution scRNA-seq datasets and minority cell types, which are more difficult to distinguish, raising concerns about the reliability of their results. To overcome these limitations, we propose a novel feature selection framework for scRNA-seq data called Mcadet. Mcadet integrates Multiple Correspondence Analysis (MCA), graph-based community detection, and a novel statistical testing approach. To assess the effectiveness of Mcadet, we conducted extensive evaluations using both simulated and real-world data, employing unbiased metrics for comparison. Our results demonstrate the superior performance of Mcadet in the selection of HVGs in scenarios involving fine-resolution scRNA-seq datasets and datasets containing minority cell populations. Overall, we demonstrate that Mcadet enhances the reliability of selected HVGs, although the impact of HVG selection on various downstream analyses varies and needs to be further investigated.

Integrating Graph Structures into Kernel Regression Models

Kernel methods are highly flexible and powerful tools for capturing complex, nonlinear relationships in data. In this paper, we propose a substantial extension of existing network-based regression models by integrating kernel methods with graph-theoretic constraints. Our approach builds upon the foundational work of Li et al. [1], who incorporated network cohesion into generalized linear models (GLMs). We extend their framework by introducing a kernelized regression model that allows for the modeling of nonlinear interactions while leveraging network data. This kernelized framework relaxes the linearity constraints of GLMs, making the model more versatile and capable of capturing complex patterns in high-dimensional spaces. We demonstrate the effectiveness of our approach using simulated and real-world datasets, such as the Teenage Friends and Lifestyle Study. Results show that our kernelized network regression model not only outperforms traditional linear and generalized linear models in predictive accuracy but also scales efficiently to larger datasets. Our work represents a significant advancement in the modeling of network-linked data, providing a robust, scalable, and interpretable framework that extends the application of kernel methods beyond their traditional constraints. Future directions include exploring more complex graph structures, such as weighted and directed graphs, and developing optimized algorithms for even larger datasets.

Empowering legal justice with AI: A reinforcement learning SAC-VAE framework for advanced legal text summarization.

Automated summarization of legal texts poses a significant challenge due to the complex and specialized nature of legal documentation. Despite the recent progress in reinforcement learning for natural language text summarization, its application in the legal domain has been less effective. This paper introduces SAC-VAE, a novel reinforcement learning framework specifically designed for legal text summarization. We leverage a Variational Autoencoder (VAE) to condense the high-dimensional state space into a more manageable lower-dimensional feature space. These compressed features are subsequently utilized by the Soft Actor-Critic (SAC) algorithm for policy learning, facilitating the automated generation of summaries from legal texts. Through comprehensive experimentation, we have empirically demonstrated the effectiveness and superior performance of the SAC-VAE framework in legal text summarization.

The Application of a Random Forest Classifier to ToF-SIMS Imaging Data.

Time-of-flight secondary ion mass spectrometry (ToF-SIMS) imaging is a potent analytical tool that provides spatially resolved chemical information on surfaces at the microscale. However, the hyperspectral nature of ToF-SIMS datasets can be challenging to analyze and interpret. Both supervised and unsupervised machine learning (ML) approaches are increasingly useful to help analyze ToF-SIMS data. Random Forest (RF) has emerged as a robust and powerful algorithm for processing mass spectrometry data. This machine learning approach offers several advantages, including accommodating nonlinear relationships, robustness to outliers in the data, managing the high-dimensional feature space, and mitigating the risk of overfitting. The application of RF to ToF-SIMS imaging facilitates the classification of complex chemical compositions and the identification of features contributing to these classifications. This tutorial aims to assist nonexperts in either machine learning or ToF-SIMS to apply Random Forest to complex ToF-SIMS datasets.

Hyperdimensional computing: a framework for stochastic computation and symbolic AI

Hyperdimensional Computing (HDC), also known as Vector Symbolic Architectures (VSA), is a neuro-inspired computing framework that exploits high-dimensional random vector spaces. HDC uses extremely parallelizable arithmetic to provide computational solutions that balance accuracy, efficiency and robustness. The majority of current HDC research focuses on the learning capabilities of these high-dimensional spaces. However, a tangential research direction investigates the properties of these high-dimensional spaces more generally as a probabilistic model for computation. In this manuscript, we provide an approachable, yet thorough, survey of the components of HDC. To highlight the dual use of HDC, we provide an in-depth analysis of two vastly different applications. The first uses HDC in a learning setting to classify graphs. Graphs are among the most important forms of information representation, and graph learning in IoT and sensor networks introduces challenges because of the limited compute capabilities. Compared to the state-of-the-art Graph Neural Networks, our proposed method achieves comparable accuracy, while training and inference times are on average 14.6× and 2.0× faster, respectively. Secondly, we analyse a dynamic hash table that uses a novel hypervector type called circular-hypervectors to map requests to a dynamic set of resources. The proposed hyperdimensional hashing method has the efficiency to be deployed in large systems. Moreover, our approach remains unaffected by a realistic level of memory errors which causes significant mismatches for existing methods.

Bayesian grouping-Gibbs sampling estimation of high-dimensional linear model with non-sparsity

In high-dimensional linear regression models, common assumptions typically entail sparsity of regression coefficients β∈Rp. However, these assumptions may not hold when the majority, if not all, of regression coefficients are non-zeros. Statistical methods designed for sparse models may lead to substantial bias in model estimation. Therefore, this article proposes a novel Bayesian Grouping-Gibbs Sampling (BGGS) method, which departs from the common sparse assumptions in high-dimensional problems. The BGGS method leverages a grouping strategy that partitions β into distinct groups, facilitating rapid sampling in high-dimensional space. The grouping number (k) can be determined using the ‘Elbow plot’, which operates efficiently and is robust against the initial value. Theoretical analysis, under some regular conditions, guarantees model selection and parameter estimation consistency, and bound for the prediction error. Furthermore, three finite simulations are conducted to assess the competitive advantages of the proposed method in terms of parameter estimation and prediction accuracy. Finally, the BGGS method is applied to a financial dataset to explore its practical utility.

An Approach for Combining Clinical Judgment with Machine Learning to Inform Medical Decision Making: Analysis of Nonemergency Surgery Strategies for Acute Appendicitis in Patients with Multiple Long-Term Conditions.

Machine learning (ML) methods can identify complex patterns of treatment effect heterogeneity. However, before ML can help to personalize decision making, transparent approaches must be developed that draw on clinical judgment. We develop an approach that combines clinical judgment with ML to generate appropriate comparative effectiveness evidence for informing decision making. We motivate this approach in evaluating the effectiveness of nonemergency surgery (NES) strategies, such as antibiotic therapy, for people with acute appendicitis who have multiple long-term conditions (MLTCs) compared with emergency surgery (ES). Our 4-stage approach 1) draws on clinical judgment about which patient characteristics and morbidities modify the relative effectiveness of NES; 2) selects additional covariates from a high-dimensional covariate space (P > 500) by applying an ML approach, least absolute shrinkage and selection operator (LASSO), to large-scale administrative data (N = 24,312); 3) generates estimates of comparative effectiveness for relevant subgroups; and 4) presents evidence in a suitable form for decision making. This approach provides useful evidence for clinically relevant subgroups. We found that overall NES strategies led to increases in the mean number of days alive and out-of-hospital compared with ES, but estimates differed across subgroups, ranging from 21.2 (95% confidence interval: 1.8 to 40.5) for patients with chronic heart failure and chronic kidney disease to -10.4 (-29.8 to 9.1) for patients with cancer and hypertension. Our interactive tool for visualizing ML output allows for findings to be customized according to the specific needs of the clinical decision maker. This principled approach of combining clinical judgment with an ML approach can improve trust, relevance, and usefulness of the evidence generated for clinical decision making. Machine learning (ML) methods have many potential applications in medical decision making, but the lack of model interpretability and usability constitutes an important barrier for the wider adoption of ML evidence in practice.We develop a 4-stage approach for integrating clinical judgment into the way an ML approach is used to estimate and report comparative effectiveness.We illustrate the approach in undertaking an evaluation of nonemergency surgery (NES) strategies for acute appendicitis in patients with multiple long-term conditions and find that NES strategies lead to better outcomes compared with emergency surgery and that the effects differ across subgroups.We develop an interactive tool for visualizing the results of this study that allows findings to be customized according to the user's preferences.

Bayesian inference of initial conditions from non-linear cosmic structures using field-level emulators

ABSTRACT Analysing next-generation cosmological data requires balancing accurate modelling of non-linear gravitational structure formation and computational demands. We propose a solution by introducing a machine learning-based field-level emulator, within the Hamiltonian Monte Carlo-based Bayesian Origin Reconstruction from Galaxies (BORG) inference algorithm. Built on a V-net neural network architecture, the emulator enhances the predictions by first-order Lagrangian perturbation theory to be accurately aligned with full N-body simulations while significantly reducing evaluation time. We test its incorporation in BORG for sampling cosmic initial conditions using mock data based on non-linear large-scale structures from N-body simulations and Gaussian noise. The method efficiently and accurately explores the high-dimensional parameter space of initial conditions, fully extracting the cross-correlation information of the data field binned at a resolution of $1.95\,h^{-1}$ Mpc. Percent-level agreement with the ground truth in the power spectrum and bispectrum is achieved up to the Nyquist frequency $k_\mathrm{N} \approx 2.79h \,\, \mathrm{Mpc}^{-1}$. Posterior resimulations – using the inferred initial conditions for N-body simulations – show that the recovery of information in the initial conditions is sufficient to accurately reproduce halo properties. In particular, we show highly accurate $M_{200\mathrm{c}}$ halo mass function and stacked density profiles of haloes in different mass bins $[0.853,16]\times 10^{14}\,{\rm M}_{\odot }\,h^{-1}$. As all available cross-correlation information is extracted, we acknowledge that limitations in recovering the initial conditions stem from the noise level and data grid resolution. This is promising as it underscores the significance of accurate non-linear modelling, indicating the potential for extracting additional information at smaller scales.

An advanced quantum support vector machine for power quality disturbance detection and identification

Quantum algorithms have demonstrated extraordinary potential across numerous fields, offering significant advantages in solving practical problems. Power Quality Disturbances (PQDs) have always been a critical factor affecting the stability and safety of electrical power systems, and accurately detecting and identifying PQDs is crucial for ensuring reliable system operation. This paper explores the application of quantum algorithms in the field of power quality and proposes a novel method using Quantum Support Vector Machines (QSVM) to detect and identify PQDs, which marks the first application of QSVM in PQD analysis. The QSVM model employed involves three main stages: quantum feature mapping, quantum kernel computation, and model training. Quantum feature mapping uses quantum circuits to map classical data into a high-dimensional Hilbert space, enhancing feature separability. Quantum kernel computation calculates the inner products between features for model training. Rigorous theoretical and experimental analyses validate our approach. This method achieves a time complexity of O(N2log(N))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$O(N^{2} \\log (N))$\\end{document}, superior to classical SVM algorithms. Simulation results show high accuracy in PQDs detection, achieving a 100% detection rate and a 96.25% accuracy rate in single PQD identification. Experimental outcomes demonstrate robustness, maintaining over 87% accuracy even with increased noise levels, confirming its effectiveness in PQDs detection and identification.

Distributional hypothesis as isomorphism between word-word co-occurrence and analogical parallelograms.

Most of the modern natural language processing (NLP) techniques are based on the vector space models of language, in which each word is represented by a vector in a high dimensional space. One of the earliest successes was demonstrated by the four-term analogical reasoning task: what is to C as B is to A? The trained word vectors form "parallelograms" representing the quadruple of words in analogy. This discovery in NLP offers us insight into our understanding of human semantic representation of words via analogical reasoning. Despite successful applications of the large-scale language models, it has not been fully understood why such parallelograms emerge by learning through natural language data. As the vector space model is not optimized to form parallelograms, the key structure related to geometric shapes of word vectors is expected to be in the data, rather than the models. In the present article, we test our hypothesis that such parallelogram arrangement of word vectors readily exists in the co-occurrence statistics of language. Our approach focuses more on the data itself, and it is different from the existing theoretical approach trying to find the mechanism of parallelogram formation in the algorithms and/or vector arithmetic operations on word vectors. First, our analysis suggested that analogical reasoning is possible by decomposition of the bigram co-occurrence matrix. Second, we demonstrated the formation of a parallelepiped, a more structured geometric object than a parallelogram, by creating a small artificial corpus and its word vectors. With these results, we propose a refined form of distributional hypothesis pointing out an isomorphism between a sort of symmetry or exchangeability and word co-occurrence statistics.

Multidimensional and Nonlinear Fractional Langevin Equations: Advancing the Theory in of Anomalous Diffusion

This paper presents a comprehensive theoretical analysis of fractional Langevin equations (FLEs) and their applications in modeling anomalous diffusion processes. We extend the classical FLE framework in several significant directions to address the complexities observed in various physical, biological, and social systems. First, we derive a multidimensional extension of the FLE, providing a robust tool for studying anomalous diffusion in higher-dimensional spaces. Second, we incorporate non-Gaussian Lévy α-stable noise into the FLE, enabling the description of systems characterized by heavy-tailed distributions and extreme events. Third, we develop a nonlinear variant of the FLE to model more intricate restoring forces and potential landscapes. Finally, we analyze a generalized FLE with different orders of fractional derivatives for the inertial and friction terms, offering a flexible framework capable of capturing a wider range of anomalous diffusion phenomena. For each extension, we prove the existence and uniqueness of solutions, derive the corresponding probability distributions, and analyze the scaling behavior of the mean squared displacement in the long-time limit. Our results demonstrate how these generalized FLEs can describe various regimes of anomalous diffusion, from subdiffusive to superdiffusive behavior. This work provides a unified theoretical foundation for understanding and modeling complex diffusive processes across diverse disciplines, paving the way for more accurate descriptions of systems exhibiting memory effects, non-Gaussian statistics, and multiscale dynamics.

Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

Multi-particle quantum walks on 3D integrated photonic chip

Quantum walks provide a speed-up in computational power for various quantum algorithms and serve as inspiration for the construction of complex graph representations. Many pioneering works have been dedicated to expanding the experimental state space and the complexity of graphs. However, these experiments are mostly limited to small experimental scale, which do not reach a many-body level and fail to reflect the multi-particle quantum interference effects among non-adjacent modes. Here, we present a quantum walk with three photons on a two-dimensional triangular lattice, which is mapped to a 19 × 19 × 19 high-dimensional state space and constructs a complex graph with 6859 nodes and 45,486 edges. By utilizing the statistical signatures of the output combinations and incorporating machine learning techniques, we successfully validate the nonclassical properties of the experiment. Our implementation provides a paradigm for exponentially expanding the state space and graph complexity of quantum walks, paving the way for surmounting the classical regime in large-scale quantum simulations.