Tensor Unified Linear Comparative Analysis (TULCA)

• Tensor Unified Linear Comparative Analysis (TULCA) is a unified framework that extends classical matrix analysis to multi-way tensor data by using dedicated tensor-mode projections.
• It employs t-scalars, t-matrices, and core tensor extraction along with a generalized Rayleigh quotient to optimize discriminative and contrastive feature extraction.
• TULCA has been validated in applications like image analysis, face reconstruction, and anomaly detection, offering higher accuracy and efficient real-time updates.

Tensor Unified Linear Comparative Analysis (TULCA) is a unified family of algorithms and frameworks for extracting, comparing, and visualizing structure in high-order tensor data, generalizing classical matrix dimensionality reduction and comparative analysis to the tensor setting. TULCA simultaneously unifies discriminant analysis and contrastive learning schemes for tensor decomposition and enables the analyst to flexibly control the criteria for highlighting, suppressing, or contrasting structure across multiple groups or experimental conditions. The TULCA methodology extends the earlier Unified Linear Comparative Analysis (ULCA)—originally developed for matrices—by introducing dedicated tensor-mode projections and core tensor extraction, together with a pipeline for empirical evaluation and visual analytics (Okami et al., 26 Jul 2025, Liao et al., 2020).

1. Mathematical Foundations: T-Scalars, T-Matrices, and Tensors

The algebraic underpinning of TULCA employs higher order data representations where matrix entries themselves are numerical arrays, termed t-scalars. Each t-scalar $$a = (a_i) \in \mathbb{C}^{I_1 \times \ldots \times I_N}$$ forms a commutative ring with entrywise addition and a circular convolution product. The convolution $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$ (with all indices modulo $I_k$) supports formulations that preserve multi-way, local, and neighborhood structure (Liao et al., 2020).

Matrices with t-scalar entries are “t-matrices” $X_{TM} \in C^{D_1 \times D_2}$, supporting scalar multiplication, t-scalar multiplication, addition, convolution product (denoted “$\circ$”), and conjugate transpose. The ring structure allows the extension of linear algebraic operations to t-matrices, and by further extension, to g-tensors (group tensors) $X_{GT} \in C^{D_1 \times \cdots \times D_M}$.

TULCA algorithms operate by associating each mode (axis) of a tensor $$X \in \mathbb{R}^{n^1 \times n^2 \times \cdots \times n^N}$$ with projection matrices $M^{(n)}$ that map the corresponding data into a lower-dimensional space, yielding a compressed “core tensor” $Z$ which preserves and clarifies group-separated or contrast-enhanced variance.

2. Unified Optimization Objective and Covariance Construction

In TULCA, the analytical goal is to extract a lower-dimensional core tensor $Z$ such that it both separates classes (discriminant analysis) and highlights variance salient to particular groups or classes (contrastive learning), or any interpolation thereof (Okami et al., 26 Jul 2025).

For each mode $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$0, TULCA constructs “within-class” and “between-class” covariances:

• Within-class covariance for class $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$1:

$(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$2

where $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$3 is the mean in mode-$(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$4.

• Between-class covariance for class $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$5:

$(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$6

where $(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$7 is the global mean along mode-$(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$8.

A weighted sum of these covariances forms the numerators and denominators of a generalized Rayleigh-quotient objective:

$(a * b)_i = \sum_{j=1}^I a_{i-j+1} b_j$9

with

$I_k$0

where the weights $I_k$1, $I_k$2, $I_k$3 in $I_k$4 are analyst-adjustable parameters that determine the comparative regime: discriminant (maximize between-class), contrastive (isolate target vs. background), or blended.

Once projection matrices are computed for each mode, the core tensor $I_k$5 is produced via sequential $I_k$6-mode products,

$I_k$7

3. Relation to TSVD and Algorithmic Unification

TULCA unifies various tensor-based analysis algorithms under a common TSVD paradigm in the t-matrix algebra (Liao et al., 2020). Classical PCA, 2DPCA, and Grassmannian Component Analysis (GCA) become special cases corresponding to particular choices of data representation and projection. The process follows:

1. Choose data representation: t-vector, t-matrix, or generic g-tensor.
2. Form a Hermitian t-matrix (or operator) $I_k$8 via inner products in the t-algebra.
3. Perform a TSVD (tensor SVD): $I_k$9.
4. Project data via $X_{TM} \in C^{D_1 \times D_2}$0, scale if needed by $X_{TM} \in C^{D_1 \times D_2}$1.

This algebraic generalization is underpinned by the commutative ring and convolution structure of t-scalars, preserved under Fourier isomorphism (convolution $X_{TM} \in C^{D_1 \times D_2}$2 Hadamard product).

Algorithmic Solution Methods

TULCA admits two solution modalities:

• Iterative generalized eigen-decomposition for the Rayleigh-quotient, selecting the leading $X_{TM} \in C^{D_1 \times D_2}$3 eigenvectors for each mode.
• Manifold optimization on the Stiefel manifold (matrices with orthonormal columns).

The dominant computational costs are constructing weighted covariances and solving the eigenproblems, with storage and time requirements comparable to traditional tensor decompositions. For interactive settings, TULCA precomputes class-covariances and rapidly recomputes eigen-solutions upon analyst-driven weight changes (Okami et al., 26 Jul 2025).

4. Practical Applications and Empirical Evaluation

TULCA has demonstrated practical efficacy in several empirical scenarios:

• Low-rank approximation and image analysis: TSVD and THOSVD outperform classical SVD and HOSVD in approximating structured image data (e.g., “Lena”), with up to 30–40 dB higher PSNR in full-tensor comparisons (Liao et al., 2020).
• Face reconstruction: TPCA and T2DPCA provide improved PSNR and lower variance for reconstruction tasks compared to standard PCA and 2DPCA.
• Supervised classification: On hyperspectral data, TGCA+NN achieves the highest overall accuracy and Cohen's κ, surpassing conventional alternatives including PCA, LDA, and GCA.
• Supercomputer log analytics: TULCA identified operational anomalies and temporal regime shifts in daily rack-temperature tensors, isolating key physical sensor variables through projection matrix analysis and interactive re-weighting (Okami et al., 26 Jul 2025).
• Mobile-health multivariate time-series: Discriminant and contrastive settings in TULCA separated activity classes (e.g., jogging vs. running), attributing separation to specific sensor channels as visualized in projection matrices.

Performance evaluations reveal that TULCA's total runtime is similar to a standard Tucker decomposition, and updates on new weights are sub-second even for tensors with millions of elements, allowing real-time “steering” by the analyst.

Data set	Size	Tucker	TULCA-all	TULCA-update
US Air quality	53×55×5	0.005 s	0.008 s	0.001 s
High-school contact	180×30×9	0.009 s	0.006 s	0.001 s
MHEALTH	480×10×23	0.018 s	0.005 s	0.001 s
K-computer logs	1086×864×4	3.849 s	11.389 s	0.573 s

5. Visualization and Interactive Steering

The dimensionality of the core tensor $X_{TM} \in C^{D_1 \times D_2}$4 poses challenges for interpretation. TULCA addresses this through CP decomposition,

$X_{TM} \in C^{D_1 \times D_2}$5

displaying:

• For mode-1: a 2D scatterplot ($X_{TM} \in C^{D_1 \times D_2}$6) visualizes group/slice structure.
• For each mode $X_{TM} \in C^{D_1 \times D_2}$7: two bar charts (one per CP-component) show contributions per dimension.

The visual analytics interface, implemented using Plotly-Dash and embeddable in Jupyter, exposes:

• A parameter panel for mode selection, dimensionality, and weight tuning.
• Brushed/core tensor views (scatterplot and linked bar charts).
• Projection-matrix heatmaps for component loadings.
• Contextual presentation of original data (e.g., time-series, spatial heat maps) synchronized with the selected scatter regions.

This modular, interactive design enables rapid hypothesis testing, expert validation of discovered patterns, and identification of outliers, salient variables, and interpretable axes of variance (Okami et al., 26 Jul 2025).

6. Theoretical and Practical Advantages

TULCA’s algebraic and optimization-based generalization provides several key advantages:

• Multi-way structure preservation: The convolution ring formalism maintains local and neighborhood correlations without flattening, capturing both spatial and spectral relationships.
• Unified framework: TSVD underlies all principal subfamilies (PCA, HOSVD, 2DPCA, GCA), enabling cross-method comparative analysis without notational or computational fragmentation.
• Dimensionality control and parallelizability: Slicewise Fourier-domain SVDs and separable mode-wise optimization allow scalable analysis.
• Empirical performance: Across approximation, reconstruction, and supervised discrimination, TULCA methods consistently yield higher accuracy, PSNR, and sensitivity to phenomena of interest compared to standard matrix or tensor approaches (Liao et al., 2020).

This suggests a plausible broader applicability for any domain in which structured, multi-group, or contrastive tensor comparisons inform expert analysis and inference.

7. Connections and Extensions

TULCA arises as a direct extension of ULCA for matrices to the $X_{TM} \in C^{D_1 \times D_2}$8-way tensor setting, by formulating and solving $X_{TM} \in C^{D_1 \times D_2}$9 simultaneous trace-ratio problems and exploiting well-defined t-algebraic generalizations of linear algebraic concepts. While current implementations focus on discriminant and contrastive paradigms, the algebraic and optimization foundations permit further integration with supervised, unsupervised, and semi-supervised tensor decomposition regimes. In addition, the core tensor visualization architecture supports seamless integration into domain-specific visual analytics workflows for exploratory and operational decision-making (Okami et al., 26 Jul 2025, Liao et al., 2020).