Efficient In-Place Tensor Rotation

• In-place tensor rotation is a technique for transforming tensors by applying cyclic shifts or orthogonal rotations without additional memory allocation.
• It employs algorithms like the 2^n+1 reversal method and mode-wise Givens rotations to achieve convergence with optimal resource usage.
• This approach is critical in scientific computing and neural architectures as it preserves symmetry and enables efficient tensor diagonalization.

In-place tensor rotation refers to the algorithmic manipulation of multi-dimensional arrays (tensors) such that their contents are rotated, shifted, or transformed via orthogonal or equivariant operations without allocating additional space for a duplicate tensor. This is a central concern in scientific computing, numerical algorithms for tensor decompositions, and machine learning applications requiring memory efficiency and symmetry preservation. Recent research formalizes rigorous in-place rotation schemes, including cyclic shifts via generalized reversal and structured orthogonal transforms, with established convergence and complexity guarantees.

1. Formal Definitions and Mathematical Framework

Let $$T \in \mathbb{R}^{d_0 \times \cdots \times d_{n-1}}$$ denote an $n$-dimensional tensor. An in-place rotation of $T$ is an operation that overwrites the elements of $T$ in memory, effecting a global transformation according to a prescribed group action (e.g., multidimensional cyclic shift, orthogonal rotation, or mode-wise Givens rotations) without auxiliary memory or only with $O(n)$ auxiliary storage, where $n$ is the tensor order.

Two central forms are prevalent:

• Cyclic Shift (Rotation): Specified by $k_{\ell} \in \mathbb{Z}/d_{\ell} \mathbb{Z}$ for each axis, the content is shifted so that

$$T'(\vec{j}) = T\left( (j_0 - k_0)\!\!\!\mod d_0, \ldots, (j_{n-1} - k_{n-1})\!\!\!\mod d_{n-1} \right)$$

Crucially, the 2ⁿ+1 reversal algorithm implements this in true $O(1)$ auxiliary space for any order $n$ (Chen, 27 Nov 2025).

• Orthogonal Rotation (Mode-Wise): For order-n symmetric tensors and $R \in SO(d)$, apply $R$ in each mode:

$$(R \cdot T)_{i_1 \ldots i_n} = \sum_{j_1,\ldots, j_n} R_{i_1 j_1} \cdots R_{i_n j_n} T_{j_1 \ldots j_n}$$

This is used in rotationally-equivariant neural architectures and for structured diagonalization (Gao et al., 2020).

2. In-Place Cyclic Shift via the $2^n+1$ Reversal Algorithm

The $2^n+1$ reversal method generalizes the classic three-step reversal for 1D arrays to cyclically shift any $n$-dimensional tensor in-place. The process comprises:

1. Global reversal: Reverse the entire tensor in lexicographically indexed storage.
2. Block reversals: Partition the tensor into $2^n$ hyperrectangular blocks according to the shift vector $k_{\ell}$, then reverse each block independently.
3. Correctness: Each element is mapped to its rotated position after the global and local block reversals, leveraging the involutive nature of reversal.

The following table concisely records the number of reversals needed by tensor order:

Order $n$	Reversals Required	Special Cases
1	3	Classic array
2	5	Matrix
3	9	Cube/tensor
n	$2^n+1$	General n-D tensor

This scheme achieves $O(N)$ time and $O(1)$ auxiliary space, where $N = \prod_{\ell=0}^{n-1} d_\ell$ (Chen, 27 Nov 2025).

3. Structured Orthogonal In-Place Tensor Rotations

In applications requiring higher-order tensor diagonalization or symmetry preservation, in-place mode-wise orthogonal rotations are critical. The Jacobi-type method for approximate orthogonal diagonalization operates as follows (Begovic, 2021):

• Pivot Selection: Select a $2 \times 2 \times 2$ subtensor indexed by a pivot pair $(p,q)$; restrict attention to this subtensor for each microstep.
• Givens Rotations per Mode: Alternately in each mode, optimize a rotation angle $\varphi$ such that the sum of squares of the two main diagonal entries of the subtensor (after rotation) is maximized. Each rotation is parametrized via the tangent-of-double-angle formula:

$$\tan(2\varphi) = \frac{2(ab-cd)}{a^2 + d^2 - c^2 - b^2}$$

(with appropriate substitutions for each mode).

• In-Place Mode-Products: Only the eight entries of the affected $2 \times 2 \times 2$ block are overwritten; the update for mode 1, for instance, is:

$$\begin{align*} A(p, :, :) &\leftarrow \cos\varphi \cdot A(p, :, :) + \sin\varphi \cdot A(q, :, :) \ A(q, :, :) &\leftarrow -\sin\varphi \cdot A(p, :, :) + \cos\varphi \cdot A(q, :, :) \end{align*}$$

with analogous updates in other modes. No full auxiliary copy is made; memory is strictly limited to the entries being updated.

This Jacobi scheme converges globally to stationary points of the objective (sum of squared diagonal entries) for general tensors, assuming a pivot strategy ensuring the existence of a sufficient rotation direction (Begovic, 2021).

4. In-Place Rotation in Equivariant Neural Architectures

Rotation-equivariance is a fundamental property for physical simulation models operating on symmetric tensors (e.g., in turbulence modeling or elasticity). RotEqNet ensures SO(n)-equivariance by:

• Standardization: Contracting the tensor to extract rotational information, aligning it to a canonical "standard position" using eigen- or QR-decomposition (depending on parity).
• Mode-wise In-place Rotation: Applies the rotation $R$ across all modes, updating in place via one-dimensional streaming per mode. This is implemented as:
  1. For each mode $m$, iterate across all fibers (subarrays when mode-$m$ index is fixed).
  2. For each fiber, compute $v \leftarrow R v$ for the fiber vector $v$.
  3. Write updated $v$ back in-place.

Only a single $n$-element buffer is required; all tensor elements are updated by iterating over the appropriate subarrays (Gao et al., 2020). This ensures both computational and memory efficiency, with complexity dominated by the total entry count and number of modes.

5. Complexity, Correctness, and Limitations

Time and space complexities of key in-place rotation schemes are as follows:

• $2^n+1$ reversal algorithm: $O(N)$ time, $O(1)$ space w.r.t. $N$; each swap and reversal is lex-order streaming and can be parallelized per block (Chen, 27 Nov 2025).

• Mode-wise orthogonal in-place rotations: $O(\prod_{\ell} d_\ell)$ flops per full sweep (neglecting small overhead per rotation); $O(n)$ buffer for per-mode multiplication.
• Correctness: Both cyclic and mode-wise rotations are involutions or elements of an orthogonal group; proofs in (Chen, 27 Nov 2025) and (Gao et al., 2020) establish correctness and structure preservation.

Edge cases such as zero shifts, full-length blocks, and highly degenerate/symmetric input tensors are handled by trivial branch conditions, often skipping unnecessary work. For antisymmetric tensors, diagonalization schemes degenerate and must be preconditioned (Begovic, 2021).

6. Applications and Extensions

In-place tensor rotation underpins:

• Approximate orthogonal tensor diagonalization and decomposition, accelerating memory-bound Jacobi-type methods in numerical multilinear algebra (Begovic, 2021).
• Rotation-equivariant neural architectures for scientific machine learning and fluid simulation, guaranteeing learned models respect geometric symmetries (Gao et al., 2020).
• Efficient cyclic shifting, streaming, and tiling of high-dimensional data in scientific computing pipelines (Chen, 27 Nov 2025).
• Generalization to higher-order and higher-rank tensors: Both reversal-based and mode-wise in-place rotations extend inductively to $n$-dimensional and $d$-mode tensors, subject to analogous correctness and performance properties.

A plausible implication is that further system-level optimization (e.g., cache-aware looping, hardware vectorization) could leverage the independence and partitioning properties inherent to these in-place algorithms to maximize performance for extremely large tensors. Additionally, since reversals and Givens rotations are composable, intricate rotation schedules for compound transformations may be synthesized via compositions of these primitives.

7. Comparative Features and Summary Table

The comparative landscape of in-place tensor rotation algorithms may be summarized as:

Algorithm/Method	Structure Preserved	Space	Time	Typical Use Case
$2^n+1$ reversal (Chen, 27 Nov 2025)	Cyclic shift/order	$O(1)$	$O(N)$	Reindexing, shifting, streaming
In-place Jacobi (Begovic, 2021)	Orthogonal/diagonal	$O(1)$	$O(N)$+	Diagonalization, multilinear algebra
Mode-wise SO(n) (Gao et al., 2020)	Symmetry/equivariance	$O(n)$	$O(N)$	Equivariant ML, physics

All approaches fundamentally avoid large auxiliary allocations, employ lexicographic or mode-wise streaming, and are mathematically grounded in involutive or orthogonal-group transformations.

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In-place tensor rotation is now a mature field, with rigorous algorithmic, mathematical, and implementation standards documented across multiple application domains (Begovic, 2021, Chen, 27 Nov 2025, Gao et al., 2020).