Multidimensional Data-Driven Framework

• The multidimensional data-driven framework is a computational paradigm that processes high-order tensor, graph, and multi-modal data while preserving inter-modal relationships.
• It employs methods like HOSVD, adaptive graph convolution, and empirical wavelet transforms to achieve efficient dimensionality reduction and enhanced predictive accuracy.
• Empirical results across domains such as neuroimaging and hyperspectral analysis demonstrate significant improvements in accuracy and scalability over traditional flat data approaches.

A multidimensional data-driven framework is a computational and algorithmic paradigm designed to analyze, represent, or process data possessing intrinsic multi-way (tensorial) structure through explicit exploitation of high-order structure and mode-specific dependencies. This class subsumes frameworks for learning, visualization, optimization, simulation, and interactive analysis in domains where classical, flat or vectorized methods cannot adequately capture inter-modal relationships, structure, or semantics. Modern architectures in this landscape leverage tensor decompositions, adaptive graph convolution, empirical wavelet frames, multi-view or multi-modal pipelines, and integrated visual/interactive components for end-to-end, mode-aware analytics at scale. Below, key theoretical, methodological, and empirical advances in this field are systematically surveyed, tracing prominent frameworks and their foundational properties.

1. Tensor-Based Ensemble and Decomposition Frameworks

Central to multidimensional analysis is the explicit exploitation of data organized as high-order tensors. The Tensor Ensemble Learning (TEL) framework (Kisil et al., 2018) operationalizes this by using truncated Higher Order Singular Value Decomposition (HOSVD) to factorize each sample $$\mathcal{X}\in\mathbb{R}^{I_1\times\dots\times I_N}$$ into orthonormal mode-specific factors $\{U^{(n)}\}$ and a compressed core $\mathcal{S}$. TEL then assembles an ensemble by training independent base learners on each mode-wise latent factor across the sample set and aggregates predictions via majority vote. The framework systematically avoids the destructive flattening of tensor data, preserves inter-modal correlation, and achieves substantial compression by tuning multilinear ranks $(R_1,\dots,R_N)$, yielding compact representations with orders of magnitude storage reduction relative to naive vectorization.

In empirical benchmarks using the ETH-80 visual dataset, TEL instantiated as TELVI (N=3) achieves classification accuracy up to 95%, significantly outperforming classical bagging with PCA-reduced vectors across various base learners. Telescoping compression rates are achieved without ad hoc projections; orthonormal factor matrices provide near-independent feature views for each base classifier, increasing diversity and overall ensemble performance. Noted limitations include the need for hyperparameter search over multilinear ranks and HOSVD’s computational overhead on large-scale tensors, with potential for parallelization and alternative decompositions (CPD, Block-Term, Tensor-Train) proposed for future extension (Kisil et al., 2018). Mode-respecting learning paradigms of this type are now foundational in fields ranging from neuroimaging to hyperspectral analysis.

2. Dimensionality Reduction: Multimodal and Multiphase Strategies

A distinct trajectory of multidimensional data-driven frameworks revolves around structured dimensionality reduction exploiting the algebraic and semantic axes of multiway data. MulTiDR (Fujiwara et al., 2020) is prototypical, treating time-dependent multivariate data as a tensor $\mathcal{X}\in\mathbb{R}^{T\times N\times D}$ and applying a two-stage compression pipeline: linear mode-wise reduction (e.g., PCA or LDA) along a chosen axis, followed by nonlinear embedding (UMAP/t-SNE) of the reduced matrix. This method retains mode-specific structure, facilitates interpretable mapping between temporal, instance, and variable dimensions, and, through integration of contrastive PCA, enables analysts to statistically dissect the features that most distinguish emergent clusters in the reduced representation.

Interactive, multi-view visual interfaces bind the analytic loop, enabling selection, drill-down, contrastive explanation, and tight coupling to domain metadata. Empirically, MulTiDR produces more interpretable and domain-relevant clusters than naive unfold+DR, as validated across diverse real-world time-series datasets (air quality, biosensor, network logs), and scales to billion-entry data without loss of exploratory agility. Potential directions include plugging in functional PCA or contrastive t-SNE, further enhancing mode-specific interpretability (Fujiwara et al., 2020).

3. Graph and Relational Multidimensional Frameworks

Data with inherent relational or interaction structure demand graph-theoretic multidimensional frameworks. A prominent architectural motif is represented by tightly-coupled LLM-agent and knowledge graph ecosystems (Wang et al., 17 Oct 2025). Here, unstructured raw data is parsed and attributed by LLM-driven agents, emitting structured tuples which are incrementally merged into a property graph (Neo4j). User exploration and analytics are enabled through rich visual interfaces (D3.js, RESTful subgraph expansion), while an LLM-based intelligent analysis module contextualizes on-demand graph traversals into chain-of-thought explanations or in situ summarization.

Bidirectional feedback between agent extraction, KG population, user-driven drilldown, and LLM-guided label or correction updates provides not only a data-driven but also user-in-the-loop multidimensional exploration capability. Empirically, this ecosystem achieves high entity and relation extraction F1 scores (~0.90), with a documented reduction in user latency for actionable insight and diminished LLM hallucination rates relative to baseline, static pipelines (Wang et al., 17 Oct 2025). Similar principles inform Collaboration Spotting (Agocs et al., 2017), which leverages algebraic view decompositions and interactive query operators on high-dimensional labeled graphs for scalable subsecond exploration on data with tens of millions of vertices and edges.

4. Data-Driven Learning and Analytics for Multimodal Time Series

In the setting of multidimensional long-sequence forecasting, data-driven frameworks increasingly integrate graph convolutional architectures with time series transformers to capture both temporal and inter-modal dependencies. The Adaptive Graph Convolutional Network (ADPGCN) framework (Wang, 2022) dynamically learns dimension (node) relationships via a parameterized adjacency matrix $\mathbf{A}_{\rm adp}$ based on trainable embeddings. Graph convolutional layers then inject cross-dimension context before Transformer-style (Informer) encoders and decoders, maintaining scalability ($\mathcal{O}(L\log L)$ for sequence length $L$) and achieving consistent gains (5–10% MSE and MAE reduction) across diverse real-world time series datasets. Such architectures are plug-and-play, applicable in sensor networks, energy forecasting, and traffic, with future potential in migrating to dynamic or hierarchical graph dependency learning (Wang, 2022).

5. Multidimensional Empirical Wavelet and Adaptive Signal Representations

Frameworks for adaptive, multidimensional signal analysis increasingly apply data-driven constructions in the frequency domain. The multidimensional empirical wavelet transform (EWT) (Lucas et al., 2024) generalizes the 1D EWT to arbitrary dimensions by constructing data-adaptive Fourier partitions and deforming any admissible mother wavelet through smooth diffeomorphisms to match local frequency modes. The resulting family of filters provides a tight (or near-tight) frame for $L^2(\mathbb{R}^n)$, with provable reconstruction via explicit dual construction. Numerical studies confirm near-perfect inversion in both smooth (Voronoi+Gabor) and crisp (Watershed+Shannon) frequency partitionings, so long as matching geometry is preserved. This approach is robust and nonparametric, opening applications in texture analysis, denoising, and high-dimension empirical signal processing.

6. Multidimensional Data-Driven Interfaces and Visual Analytics

Application frameworks such as idwMapper (Sarigai et al., 2024) demonstrate the effective combination of interactive, crossfiltered multi-view dashboards and web mapping for “sensing” dozens of dimensions in geospatial data at scale. In lieu of algorithmic dimensionality reduction, a coordinated-client index (Crossfilter.js) enables high-dimensional slicing, brushing, and drilldown across synchronized maps and charts, achieving sub-30 ms update rates on million-row tables entirely in-browser. This paradigm enables domain-agnostic, rapid development of web-based multidimensional visual analytics, supporting literature mining, university rankings, and cohort mapping. Notably, “idw” refers only to “Interactive Data-driven Web”, and does not denote spatial interpolation, which would constitute a future extension (Sarigai et al., 2024).

7. Current Limitations, Open Directions, and Synthesis

Multidimensional data-driven frameworks fundamentally advance analysis by (i) preserving and exploiting tensor, graph, or manifold structure, (ii) enabling flexible, scalable, and explainable mode-specific processing, and (iii) interfacing with modern learning architectures for adaptive, user-driven insight generation. Core limitations are computational—tensor factorization and large-scale graph processing remain expensive, and model selection (compression ranks, graph adjacency) is often nontrivial and data-intensive. Opportunities include integrated, online, and meta-learning frameworks for streaming multidimensional data, richer ensemble fusion (e.g., stacking outputs from heterogeneous decompositions), increased automation of multiway model selection, and further integration of causal and statistical inference within these general pipelines.

The multidimensional data-driven framework as a research and engineering paradigm thus defines a broad, theoretically motivated and empirically validated toolkit for ubiquitous problems in science and industry, where the high-dimensional, multi-relational, or temporal nature of data demands mode-aware, adaptability-centric computation (Kisil et al., 2018, Fujiwara et al., 2020, Wang, 2022, Wang et al., 17 Oct 2025, Lucas et al., 2024, Sarigai et al., 2024).