Truncated Tucker Decomposition with Error Control

• The paper introduces error-controlled truncated Tucker decomposition algorithms that guarantee near-optimal low multilinear rank approximations using methods like HOSVD and randomized sketching.
• The paper presents interpolatory CUR-type approaches that directly sample tensor fibers to preserve data structure while providing probabilistic error bounds.
• The paper further investigates randomized, streaming, and rank-adaptive strategies to enhance scalability and accuracy in high-dimensional tensor approximations.

Truncated Tucker Decomposition with Error Control

The truncated Tucker decomposition with error control targets the computation of best-approximation or near-best-approximation low multilinear rank representations of a tensor, with explicit guarantees on the Frobenius norm error. This paradigm is critical in high-dimensional settings where efficiency, rigor, and structure preservation must be reconciled, and has motivated extensive algorithmic innovation. Approaches now include randomized sketching, interpolatory CUR/Tucker, streaming updates, and parallel implementations, all equipped with rigorous a priori or probabilistic error bounds.

1. Problem Formulation and Classical Solution

Given a $d$-way tensor $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$ and target multilinear ranks $(r_1, \dots, r_d)$, the truncated Tucker decomposition seeks orthonormal factor matrices $U_n \in \mathbb{R}^{I_n \times r_n}$ and a core tensor $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$ minimizing the Frobenius error: $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$ The classical (deterministic, optimal) approach is the truncated higher-order singular value decomposition (HOSVD), which first computes the mode-$n$ unfoldings $X_{(n)}$, extracts the leading $r_n$ left singular vectors for each mode as $U_n$, and forms the core via mode products. The approximation error satisfies

$$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$0

where $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$1 are the singular values of mode-$$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$2 unfoldings. This forms the baseline against which all efficient and randomized truncation algorithms with error control are measured (Cao et al., 2022).

2. Interpolatory and CUR-Type Tucker Decompositions

To preserve data structure such as sparsity or nonnegativity, interpolatory or CUR-type Tucker decompositions select columns (or fibers) directly from the original tensor using sampling methods. A leading strategy is the L-DEIM (Leverage-augmented Discrete Empirical Interpolation Method), which operates as follows:

• For each mode $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$3: (1) Unfold $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$4 along mode $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$5; (2) Approximate singular value decomposition (SVD) to extract $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$6 right singular vectors; (3) Select $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$7 columns using L-DEIM—first selecting $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$8 indices via classic DEIM, then augmenting with additional columns by leverage-score ranking.
• Assemble the indices into sampled fiber sets $$\mathcal{X} \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_d}$$9, retrieve the corresponding columns to form $(r_1, \dots, r_d)$0.
• Compute the core by

$(r_1, \dots, r_d)$1

where $(r_1, \dots, r_d)$2 denotes the Moore–Penrose pseudoinverse.

Error control is realized by probabilistic bounds (for the randomized L-DEIM–HOID variant) parameterized by an oversampling parameter $(r_1, \dots, r_d)$3, SVD quality parameters $(r_1, \dots, r_d)$4, $(r_1, \dots, r_d)$5, and the singular value decay. Explicitly, for oversampling $(r_1, \dots, r_d)$6, error satisfies

$(r_1, \dots, r_d)$7

with failure probability controlled by $(r_1, \dots, r_d)$8 (typically set so that the failure rate is $(r_1, \dots, r_d)$9 in practice) (Cao et al., 2022).

These structure-preserving decompositions are particularly advantageous when factor interpretability or data format adherence is necessary.

3. Randomized and Streaming Algorithms with Error Guarantees

Randomized algorithms dramatically reduce the computational cost of Tucker truncation and inherently facilitate error estimation via probabilistic analysis. Core methods include:

• Randomized T-HOSVD/ST-HOSVD with power iteration and/or adaptive shifts: Each mode is sketched with a Gaussian random projection (optionally followed by power iterations and adaptive shift in the power method to enhance spectral gap), and an orthonormal basis is extracted (Che et al., 2023, Che et al., 5 Jun 2025). With oversampling $U_n \in \mathbb{R}^{I_n \times r_n}$0, power parameter $U_n \in \mathbb{R}^{I_n \times r_n}$1, and shift parameter $U_n \in \mathbb{R}^{I_n \times r_n}$2, error is bounded (with high probability) in terms of higher moments of the singular value tail and decays rapidly with $U_n \in \mathbb{R}^{I_n \times r_n}$3 and $U_n \in \mathbb{R}^{I_n \times r_n}$4:

$U_n \in \mathbb{R}^{I_n \times r_n}$5

where $U_n \in \mathbb{R}^{I_n \times r_n}$6 decay with $U_n \in \mathbb{R}^{I_n \times r_n}$7 and shifting makes $U_n \in \mathbb{R}^{I_n \times r_n}$8 often sufficient (Che et al., 5 Jun 2025).

• Oblivious Subspace Embeddings (OSEs) for compressed HOOI: Each mode is compressed by a JL-optimal embedding, with embedding dimension $U_n \in \mathbb{R}^{I_n \times r_n}$9. The HOOI iterates then operate in the compressed domain, with relative error guarantees:

$$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$0

enabling rigorous $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$1-approximate solutions with substantial time and memory savings (Pietrosanu et al., 2024).

• Streaming Adaptive ST-HOSVD: For tensors growing along a streaming mode (e.g., time), factor matrices and the core are updated incrementally in one pass without storing the entire history. Component ranks and SVD tolerances are dynamically chosen to keep the total Frobenius error below a user-specified $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$2 at all times (De et al., 2023).

These approaches deliver near-optimal error, sharp time-memory tradeoffs, and adaptivity to streaming and parallel hardware.

4. Advanced Error-Control Strategies and Rank Adaptivity

Rank-adaptive algorithms determine the minimum multilinear ranks needed to meet a prescribed absolute or relative error threshold, rather than require fixed ranks as input. In the rank-adaptive HOOI framework, each mode truncation rank $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$3 is dynamically set at each sweep by evaluating the tail singular-value energy against error budgets derived from the global constraint

$$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$4

Rank reduction is monotonic (non-increasing per sweep) with convergence guaranteed after finitely many updates, and local optimality follows from the Eckart–Young theorem. Empirical results show that rank-adaptive methods recover the true underlying multilinear rank in synthetic and real data, leading to significantly more compact decompositions for a given target error (Xiao et al., 2021).

Hybrid truncation methods (e.g., in ATC) further split the total error budget between Tucker truncation and quantization, delivering tight error control post-compression through careful error tracking and blockwise parallel quantization (Baert et al., 2021).

5. Mode-Parallel and Scalable Implementations

For high-order tensors or those with extremely large modes, mode-parallel randomized algorithms enable resource-efficient Tucker truncation. Subsampled randomized HOSVD (Sub-R-HOSVD) applies randomized range-finding to suitably sampled fibers from each mode independently. With sampling budget $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$5 (where $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$6 denotes the coherence of the mode-$$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$7 unfolding), expectations of the truncation error can be tightly controlled: $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$8 This does not require explicit formation of the mode unfoldings and is fully compatible with distributed memory and parallel compute environments (Iannacito et al., 22 Mar 2026).

Block-Krylov iterative sketching (rBKI-TK) further enhances error and noise-robustness for high-noise or low-signal data, attaining error within a low factor of the deterministic best multilinear-rank error with very low oversampling and power–iteration cost (Qiu et al., 2022).

6. Practical Recommendations and Comparative Perspectives

Empirical findings across large synthetic and real datasets support the following guidelines:

• Use oversampling $$\mathcal{G} \in \mathbb{R}^{r_1 \times \cdots \times r_d}$$9–$$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$0 for randomized SVDs or OSEs per mode.
• For L-DEIM or CUR-Tucker, set the DEIM parameter $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$1.
• Adaptive-shifted power iterations with $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$2 or $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$3 suffice to closely match deterministic HOSVD accuracy.
• Rank selection can be automated via singular-value energy partitioning to meet prescribed $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$4 thresholds (Cao et al., 2022, Che et al., 5 Jun 2025, Xiao et al., 2021).
• For streaming or online scenarios, incremental SVDs and error splitting per non-streaming mode are mandatory for time-unbounded data (De et al., 2023).

In the presence of strong prior knowledge on tensor structure (e.g., sparsity, nonnegativity), interpolatory Tucker (e.g., L-DEIM–HOID) is preferred for structure preservation. Conversely, for extremely large, distributed, or high-order data, randomized or mode-parallel approaches are recommended for scalability and efficient error control.

A summary comparison of prominent algorithmic classes is given below.

Approach	Error Guarantee	Scalability	Structure Preservation
Classical HOSVD/ST-HOSVD	Sum of tail squares: $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$5	Moderate	No
Randomized T-HOSVD/ST-HOSVD	Probabilistic: $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$6	High	No
L-DEIM–CUR-HOID	Multiplicative factor on tail singular values	High	Yes (fibers/columns chosen)
OSE-compressed HOOI	Relative $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$7 of best rank approximation	Very High	No
Block Krylov (rBKI-TK)	$$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$8 times best (deterministic) error	Very High	No
Streaming ST-HOSVD	Global $$\min_{U_n,\, \mathcal{G}} \|\mathcal{X} - \mathcal{G} \times_1 U_1 \times_2 \cdots \times_d U_d\|_F$$9 enforced slice-by-slice	Highest	No

7. Limitations and Ongoing Directions

Limitations of current paradigm variants include:

• Multiplicative error growth factors or additional tail-energy terms in interpolatory or randomized decompositions versus optimal HOSVD.
• Non-optimality of random sketching under extreme coherence unless sampling budgets are increased.
• Structured random embeddings (e.g., SRFT, CountSketch with adaptive shift) lack complete theoretical development in the context of Tucker error control (Che et al., 5 Jun 2025).
• Block-Krylov and fiber-sampling techniques are sensitive to rank overestimation in low-signal regimes.
• Multi-shift or Rayleigh-quotient adaptive strategies may further accelerate randomized computations, but their full analysis remains open.

Practical rank-adaptive stopping and distributed parallelization strategies are active areas of development, as is generalization to non-Euclidean domains (e.g., tensors with graph structure or in nonstandard data types).

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References:

(Cao et al., 2022) – CUR-type Tucker via L-DEIM with error bounds (Che et al., 2023) – Randomized/approximate T-HOSVD and error analysis (Che et al., 5 Jun 2025) – Randomized (S)T-HOSVD with adaptive shift power scheme (Pietrosanu et al., 2024) – OSE-compressed HOOI with JL-type error control (Qiu et al., 2022) – rBKI-TK, randomized block Krylov with rigorous bounds (Baert et al., 2021) – Hybrid (ST-HOSVD + quantization) and error tracking (Xiao et al., 2021) – Rank-adaptive HOOI, local optimality and convergence (De et al., 2023) – Streaming, error-controlled Tucker update (Iannacito et al., 22 Mar 2026) – Mode-parallel randomized Tucker with sampling bounds