{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a dense tensor feature basis with a truncated star_M-SVD basis computed after projection into the bilateral-reflection invariant subspace. The layer stores only the independent half of each symmetric basis slice, while reconstructing the full basis on the fly; this should reduce parameters and matrix multiplications without discarding the symmetric signal. The construction can be used as a frozen compression layer, a trainable low-rank adapter, or an initialization for a convolutional feature basis.",
 "formulas_latex": [
  "$$\\mathcal{A}=\\mathcal{U}\\star_{\\mathbf{M}}\\mathcal{S}\\star_{\\mathbf{M}}\\mathcal{V}^{\\mathsf{T}}=\\sum_{i=1}^{r}\\mathcal{U}_{:,i,:}\\star_{\\mathbf{M}}\\mathcal{S}_{i,i,:}\\star_{\\mathbf{M}}\\mathcal{V}_{:,i,:}^{\\mathsf{T}}.$$",
  "$$\\widehat A^{(k)}=\\widehat U^{(k)}\\widehat S^{(k)}(\\widehat V^{(k)})^{\\mathsf T},\\qquad \\widehat{\\mathcal A}=\\mathcal A\\times_3\\mathbf M,\\qquad \\mathcal U=\\widehat{\\mathcal U}\\times_3\\mathbf M^{-1}.$$",
  "$$\\lVert\\mathcal B\\rVert_F^2=\\frac{1}{c^2}\\sum_k\\lVert\\widehat B^{(k)}\\rVert_F^2.$$",
  "$$P_m(X)=\\frac12(X+J_mX),\\qquad \\widehat A^{(k)}_{\\mathrm{sym}}=P_m(\\widehat A^{(k)}),\\qquad \\widehat A^{(k)}_r=\\widehat U^{(k)}_{:,1:r}\\widehat S^{(k)}_{1:r,1:r}(\\widehat V^{(k)}_{:,1:r})^{\\mathsf T}.$$"
 ],
 "id": 149,
 "implementation": "Integrate this at a tensor feature bottleneck in a CNN or video encoder, before a classifier or before a costly spatial-mixing block. Let the feature tensor be \\(A\\in\\mathbb R^{m\\times p\\times n}\\), where \\(m\\) is the reflected spatial coordinate, \\(p\\) is channel or sample width, and \\(n\\) is the temporal, frequency, or feature-mode length. Choose an invertible mode transform \\(M\\), such as a DCT or FFT, and compute `Ahat = mode3_transform(A, M)`. For every frontal slice `k`, construct the reflection permutation matrix `J`, compute `Asym = 0.5 * (Ahat[k] + J @ Ahat[k])`, and run a rank-`r` truncated SVD `U,S,Vh = svd(Asym, rank=r)`. Enforce numerical symmetry with `U = 0.5 * (U + J @ U)` and re-orthogonalize with QR if necessary. Store only the independent upper half of `U`; in the forward pass reconstruct it as `U_full = concat(U_top, reverse(U_top))`, treating the center row separately when `m` is odd. Compute `Yhat[k] = U_full @ diag(S) @ Vh`, then apply the inverse mode transform. For a trainable low-rank adapter, initialize these factors from pretrained activations and optimize only `U_top`, `S`, and `V`; after each optimizer step project `U` with `P_m` and re-orthogonalize. The mathematical quantities are the transform-domain SVD, reflection projector, and Frobenius scaling. Estimate rank empirically using the retained-energy ratio `sum(S[:r]^2) / sum(S^2)`. Start with Fashion-MNIST, aligned CelebA faces, or a symmetric-object dataset, using ResNet-18 with a dense bottleneck as baseline. Compare accuracy, reconstruction error, forward FLOPs, and basis storage against ordinary low-rank SVD and an uncompressed layer. Success is equal accuracy at 2x-5x lower basis storage or lower latency at equal accuracy.",
 "math_summary": "The paper defines the star_M-SVD for \\(\\mathcal A\\in\\mathbb R^{m\\times p\\times n}\\) by \\(\\mathcal A=\\mathcal U\\star_{\\mathbf M}\\mathcal S\\star_{\\mathbf M}\\mathcal V^{\\mathsf T}=\\sum_{i=1}^{r}\\mathcal U_{:,i,:}\\star_{\\mathbf M}\\mathcal S_{i,i,:}\\star_{\\mathbf M}\\mathcal V_{:,i,:}^{\\mathsf T}\\), where \\(\\mathcal U\\) and \\(\\mathcal V\\) are star_M-orthogonal, \\(\\mathcal S\\) is f-diagonal, and \\(r\\leq\\min(m,p)\\) is the number of nonzero singular tubes. The transform-domain implementation forms \\(\\widehat{\\mathcal A}=\\mathcal A\\times_3\\mathbf M\\), computes ordinary matrix SVDs \\(\\widehat A^{(k)}=\\widehat U^{(k)}\\widehat S^{(k)}(\\widehat V^{(k)})^{\\mathsf T}\\) independently for every frontal slice \\(k\\), then applies \\(\\times_3\\mathbf M^{-1}\\). The Frobenius normalization is \\(\\lVert\\mathcal B\\rVert_F^2=c^{-2}\\sum_k\\lVert\\widehat B^{(k)}\\rVert_F^2\\), where \\(c\\) is the transform normalization constant. Let \\(J_m\\in\\mathbb R^{m\\times m}\\) reverse the reflected spatial coordinate, \\((J_mx)_i=x_{m+1-i}\\). The invariant projector is \\(P_m(X)=\\tfrac12(X+J_mX)\\). Apply it to each transformed frontal slice before SVD. A symmetric left basis satisfies \\(J_m\\widehat U^{(k)}=\\widehat U^{(k)}\\), so only its top \\(\\lceil m/2\\rceil\\) rows need to be stored; the remaining rows are generated by reflection.",
 "math_tags": [
  "tensor-decomposition",
  "linear-algebra",
  "representation-theory",
  "numerical-analysis"
 ],
 "ml_areas": [
  "cnn",
  "embedding",
  "fine-tuning",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.24985",
  "arxiv_url": "https://arxiv.org/abs/2608.24985",
  "summary_what_math_gives_to_ml": "The paper gives a constructive tensor factorization that preserves bilateral reflection symmetry while retaining the matrix-SVD-like decomposition and Frobenius error behavior. Its transferable asset is the ability to constrain a learned low-rank basis to a symmetry subspace and remove redundant parameters without flattening spatial or temporal modes. A practical neural-network adaptation is a symmetry-preserving low-rank feature layer or adapter: transform a tensor along its third mode, project each frontal slice into the reflection-invariant subspace, compute truncated SVD factors, and reconstruct the full basis on demand. This is most promising for image, video, medical-imaging, and object-recognition models with bilateral symmetry.",
  "title": "A Symmetry-Preserving Tensor $\\star_{\\mathbf{M}}$-SVD",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "memory",
  "speedup",
  "accuracy"
 ],
 "title": "Reflection-Preserving Tensor Low-Rank Layer",
 "url": "https://synthcore.org/idea/149/reflection-preserving-tensor-low-rank-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}