Hierarchical Tucker Tensor Format

• Hierarchical Tucker Tensor (HTT) is a tree-structured format that decomposes high-dimensional tensors recursively to drastically reduce storage and computational complexity.
• It leverages a binary dimension tree and transfer tensors to enable efficient low-rank approximations via methods like iterative hard thresholding and Riemannian optimization.
• The HTT approach underpins practical applications in tensor completion, scientific computing, and machine learning while inspiring research on adaptive and randomized decomposition strategies.

The Hierarchical Tucker Tensor (HTT) format is a tree-structured, multi-level extension of the Tucker decomposition designed to enable efficient representation and computation with high-order, potentially very high-dimensional tensors. By recursively decomposing a tensor into nested subspaces associated with nodes of a dimension tree, the HTT format realizes drastic reductions in storage and computational complexity compared to conventional dense or even classical Tucker models. This format supports practical algorithms for tensor completion, low-rank approximation, scientific computing, and machine learning, and its mathematical foundations encompass differential geometry, optimization on manifolds, and the theory of tensor networks.

1. Algebraic and Geometric Structure of the HTT Format

The HTT format generalizes the Tucker model by introducing a dimension partition (binary) tree $T$ over the $d$ tensor modes. Every node $\alpha \subset D = \{1, \ldots, d\}$ of $T$ represents a subspace $U_\alpha$, and edges encode recursive two-way tensor product factorizations. For a node $\alpha$ with children $\alpha_1$, $\alpha_2$, the local representation is

$$b^\alpha_\ell = \sum_{i=1}^{r_{\alpha_1}}\sum_{j=1}^{r_{\alpha_2}} b^{\alpha}(i,j,\ell) \cdot b^{\alpha_1}_i \otimes b^{\alpha_2}_j, \qquad \ell = 1, \ldots, r_\alpha,$$

where $b^\alpha$ is a "transfer tensor" that fuses the lower-level factors. The process recurses from the leaves (modes) to the root, culminating in a full factorization of the original tensor. The data complexity of the HTT format is

$O(n \cdot d \cdot r + d \cdot r^3),$

with $n = \max n_i$, $r = \max r_\alpha$. This scaling is only linear in the order $d$ and polynomial in $r$, contrasting sharply with the $O(n^d)$ cost of an explicit array representation.

A notable special case is the Tensor Train (TT) format, corresponding to an unbalanced "train-track" tree (Buczyńska et al., 2015). In the TT representation, the tensor is written as a contracted product of three-way core tensors, yielding further computational simplifications but less flexibility for certain tensor structures.

2. Tree-Based Rank, Minimal Subspaces, and Manifold Geometry

Each edge or node $\alpha$ in $T$ is characterized by a hierarchical (tree-based) rank $r_\alpha = \dim U_\alpha$, which equals the rank of the corresponding matricization of the tensor with respect to the index split induced by $\alpha$. The collection of ranks for all nodes encodes the complexity and approximation capacity of the HTT representation (Falco et al., 2018).

The set of tensors with fixed tree-based ranks forms an analytic Banach manifold (Falco et al., 2015, Falco et al., 2018). Local coordinate charts are constructed using the Grassmannians of minimal subspaces at every node and the associated coefficient (core) tensors. This fibre bundle structure facilitates the use of differential calculus on the manifold, supports the explicit calculation of tangent spaces, and underpins algorithms for low-rank approximation and model reduction.

Furthermore, the manifold of HTT tensors with fixed tree-based ranks is itself the intersection of Tucker manifolds associated with different partitions (levels) of the tree (Falcó et al., 2020).

3. Algorithmic Approaches and Optimization on the HTT Manifold

Algorithms exploiting the HTT structure address both tensor completion from incomplete measurements and general low-rank recovery problems. Typical strategies include:

• Iterative Hard Thresholding (IHT): Alternates between a gradient step in the ambient tensor space and projection onto the low-rank manifold via Hierarchical SVD (HSVD) with truncation (Rauhut et al., 2014). The projection step yields a quasi-optimal approximation with controllable error bounds.
• Riemannian Optimization: Leverages the smooth manifold structure to compute Riemannian gradients, project onto tangent spaces, and retract to the manifold (typically using QR factorization-based retraction). This underpins both steepest descent and (nonlinear) conjugate gradient methods for efficient solution of optimization problems directly in the HT parameterization (Silva et al., 2014).
• Regularization and Gramian Matrices: Overfitting in highly undersampled completion problems is mitigated by introducing regularizers involving Gramian matrices $G_t$ at each node $t$, constructed recursively. Regularization terms such as $\sum_t \mathrm{tr}(G_t) + \mathrm{tr}(G_t^{-1})$ penalize both excessively large and small singular values of corresponding matricizations, thus preventing instability near the rank-deficient boundary (Silva et al., 2014).
• Distributed Algorithms and Parallelization: A balanced tree structure enables efficient parallelization, assigning each tree node to a compute node, supporting distributed contraction, orthogonalization, truncation, and operator application, with parallel runtimes scaling as $O(\log d)$ (Grasedyck et al., 2017).

4. Theoretical Properties: Restricted Isometry, Expressiveness, and Efficiency

The tensor Restricted Isometry Property (TRIP) plays a central role in theoretical guarantees for recovery from incomplete measurements. For a measurement operator $\mathcal{A}$, the TRIP of order $r$ ensures

$$(1-\delta_r)\|u\|^2 \leq \|\mathcal{A}u\|^2 \leq (1+\delta_r)\|u\|^2, \qquad \forall u \in \mathcal{M}_r,$$

where $\mathcal{M}_r$ is the manifold of HTT tensors of prescribed tree-based rank. For Gaussian measurement maps, satisfying the TRIP with high probability requires a number of measurements $m$ that scales with the number of degrees of freedom in the representation, e.g., $O(d n r^2 \log (dr))$ for TT and $O(r^d + d n r)$ for Tucker (HOSVD) (Rauhut et al., 2014).

Comparisons between HT (hierarchical) and TT (train-track) formats show that, although both are subclasses of tree tensor networks, the hierarchical format is asymptotically more efficient for generic high-order tensors. Converting an HT tensor with rank $r$ on a balanced tree to TT format may require exponentially larger TT ranks, as established by the Hackbusch conjecture: $$\mathrm{HF}(r, k) \not\subset \mathrm{TT}(r', 2^k), \quad \forall r' < r^{\lceil k/2 \rceil}$$ where $\mathrm{HF}(r, k)$ denotes hierarchical format with rank $r$ and tree depth $k$ (Buczyńska et al., 2015, Buczyńska, 2018).

This efficiency gap highlights the importance of aligning the choice of decomposition format with the structure of the target tensor.

5. Extensions, Randomization, and Advances

Recent research directions include:

• Sparse and High-Order Extensions: Sparse HTT variants (Sparse H-Tucker) enable scalable factorizations of extremely high-order, large, and sparse tensors by leveraging CUR decompositions and nested sampling, avoiding explicit formation of dense cores and yielding near-linear scaling in data sparsity (Perros et al., 2016).
• Randomized Algorithms: Randomized SVD techniques have been adapted to the TT and HT settings to accelerate decomposition, with random test matrices used to approximate the column space of matricizations at each node. Error bounds inherit a multiplicative factor depending on the tree depth and oversampling, analogous to matrix and TT cases (Huber et al., 2017).
• Incremental and Streaming Decomposition: Recent works have extended HTT to streaming and batch-incremental settings (e.g., BHT-l2r, HT-RISE) for online tensor processing without reconstructing the full dataset, achieving favorable compression and computational efficiency (Aksoy et al., 21 Dec 2024).
• Applications in Machine Learning and Scientific Computing: HTT compression improves both storage and generalization in neural networks (particularly for RNN and fully-connected layers) (Yin et al., 2020, Wu et al., 2020), outperforms TT and other decompositions for balanced high-dimensional weight tensors, and provides an optimal format for hierarchical scientific data (Silva et al., 2014, Sands et al., 27 Jun 2024).

6. Limitations, Open Problems, and Future Directions

Despite its efficiency and flexibility, the HTT format has limitations:

• Rank Selection and Model Selection: The optimal choice of tree structure and rank tuple remains application dependent and nontrivial, especially for highly anisotropic or irregular data.
• Exponentially Growing CP-Rank: For generic tensors, the minimal CP-rank (tensor rank) within the HTT manifold still grows exponentially with the number of leaves, so storage and computation may become intractable for some problems, despite the exponential reduction compared to dense arrays (Buczyńska, 2018).
• Algebraic and Geometric Complexity: The non-uniqueness of HTT representations (gauge degrees of freedom), the need for orthogonality constraints, and the complexity of projection and retraction operations complicate both theory and implementation.
• Compatibility with Downstream Tasks: Certain machine learning or inference pipelines may require flattening or contraction patterns not efficiently compatible with the fixed tree structure, prompting hybrid or adaptive decomposition strategies.

Ongoing efforts target adaptive and hybrid structures (Wu et al., 2020), further advances in randomized, distributed, and streaming algorithms, and the development of theoretical tools for understanding topological properties, optimality criteria, and regularization specifically tailored to applications in scientific computing, big data analytics, and statistical learning.

7. Summary Table: HTT vs Classical and TT Formats

Feature	Tucker	HTT (Hierarchical Tucker)	TT (Tensor Train)
Complexity (params)	$O(n^d)$	$O(n \cdot d \cdot r + d r^3)$	$O(d n r^2)$
Structure	Flat, core	Tree-based, multilevel	Chain/unbalanced tree
Expressive efficiency	Poor for $d \gg 3$	High (balanced rank for generic tensors)	Less for generic, best for chain-like tensors
Best for	Low-order	High-order, balanced or generic	High-order, chain-structured
Conversion overhead	N/A	Low for HTT$\to$TT, high for TT$\to$HTT (exponential gap)	High for TT$\to$HTT for generic tensors

This table compares the key storage and structural properties of three formats discussed in the referenced literature (Rauhut et al., 2014, Buczyńska et al., 2015, Perros et al., 2016, Yin et al., 2020).

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The Hierarchical Tucker Tensor format constitutes a central analytical and algorithmic pillar for modern high-dimensional and multiway data analysis, with a geometric, algebraic, and computational theory supporting wide-ranging applications and continued theoretical development.