{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a channelwise wavelet or strided-convolution front end with vector-valued wavelet filters that deliberately pair different scalar wavelets across channels. The resulting subbands retain compact-support multiscale structure and can be recombined exactly, while a small learned 1x1 mixing layer operates on the cross-channel coefficients instead of learning a full expensive convolution at every scale.",
 "formulas_latex": [
  "$$\\Psi^{2}_{j,k}(x,y)=\\big(\\phi_{1}(2^{j}x-k_{1})\\psi_{2}(2^{j}y-k_{2}),\\phi_{2}(2^{j}x-k_{1})\\psi_{1}(2^{j}y-k_{2})\\big)^{T}$$",
  "$$\\Psi^{3}_{j,k}(x,y)=\\big(\\psi_{1}(2^{j}x-k_{1})\\phi_{1}(2^{j}y-k_{2}),\\psi_{2}(2^{j}x-k_{1})\\phi_{2}(2^{j}y-k_{2})\\big)^{T}$$",
  "$$\\Psi^{4}_{j,k}(x,y)=\\big(\\psi_{1}(2^{j}x-k_{1})\\phi_{2}(2^{j}y-k_{2}),\\psi_{2}(2^{j}x-k_{1})\\phi_{1}(2^{j}y-k_{2})\\big)^{T}$$",
  "$$c^{r}_{j,k}=\\sum_{i=1}^{2}\\int f_i(x,y)\\,\\Psi^{r}_{j,k,i}(x,y)\\,dx\\,dy,\\qquad \\widehat f=\\sum_{r,j,k}c^{r}_{j,k}\\Psi^{r}_{j,k},\\qquad \\|f\\|_2^2=\\sum_{r,j,k}|c^{r}_{j,k}|^2$$"
 ],
 "id": 2898,
 "implementation": "1. Integration point: use this module as the first two downsampling stages of a small CNN or U-Net, replacing a stride-2 convolution. The input is a tensor f of shape [batch, 2, H, W]; for RGB, first use a learned or fixed 3-to-2 projection, or generalize the two-channel index patterns to m=3. Construct short FIR filters sampled from compactly supported φ_1, φ_2, ψ_1, ψ_2. 2. Pseudocode: `for scale j: for each listed subband r: y[r]=conv2d(f[0], h[r,0]) + conv2d(f[1], h[r,1]); y[r]=downsample2(y[r]); z[r]=pointwise_group_mix_r(y[r]); f_next=concat(z[r])`; set `h[r,i]` from the two components of Ψ^r, and use the corresponding synthesis filters for reconstruction. Include the omitted scaling-function branch as the low-pass output. 3. Mathematical quantities computed exactly are the cross-paired filter coefficients and the synthesis bank. Estimate only normalization constants empirically by checking the discrete frame Gram matrix; rescale filters so its diagonal is one and its off-diagonal Frobenius norm is small. Add an optional diagnostic `energy_error=abs(||f||²-Σ_r||y[r]||²)/||f||²`. Keep the wavelet filters frozen for the first experiment, then compare frozen filters with jointly learned filters initialized from them. 4. First experiment: train a two-channel image denoising CNN on synthetic correlated Gaussian fields or stereo patches, comparing a stride-2 convolution, a channelwise Haar/DWT front end, and this cross-channel bank at matched parameter count and FLOPs. Measure validation PSNR, reconstruction error after analysis/synthesis, and energy error. Success means higher PSNR at equal FLOPs, lower aliasing under 2x downsampling, and reconstruction error near numerical precision before any learned layers.",
 "math_summary": "The construction groups scalar tensor-product wavelets by channel-index patterns A_1={(1,1),(2,2)} and A_2={(1,2),(2,1)}. For two channels and two spatial dimensions, representative vector wavelets are Ψ^2_{j,k}(x,y)=(φ_1(2^j x-k_1)ψ_2(2^j y-k_2), φ_2(2^j x-k_1)ψ_1(2^j y-k_2))^T, Ψ^3_{j,k}(x,y)=(ψ_1(2^j x-k_1)φ_1(2^j y-k_2), ψ_2(2^j x-k_1)φ_2(2^j y-k_2))^T, and Ψ^4_{j,k}(x,y)=(ψ_1(2^j x-k_1)φ_2(2^j y-k_2), ψ_2(2^j x-k_1)φ_1(2^j y-k_2))^T, with Ψ^5 and Ψ^6 using ψ in both spatial coordinates. Here φ_1,φ_2 are scalar scaling functions, ψ_1,ψ_2 are scalar wavelets, j is the dyadic scale, k=(k_1,k_2) is the spatial translation, and T denotes transpose. The scalar functions are assumed to come from an orthonormal or Parseval multiresolution basis, so analysis followed by synthesis preserves or approximately preserves energy: ||f||_2^2=Σ_r,j,k |c^r_{j,k}|^2 for orthonormal normalization. In an ML tensor implementation, the corresponding analysis coefficient is c^r_{j,k}=Σ_{i=1}^2∫ f_i(x,y)Ψ^r_{j,k,i}(x,y)dxdy; the learned generalization is C^r_{j,k}=W_r c^r_{j,k}, where W_r is a small channel-mixing matrix.",
 "math_tags": [
  "harmonic-analysis",
  "functional-analysis",
  "linear-algebra"
 ],
 "ml_areas": [
  "cnn",
  "embedding"
 ],
 "paper": {
  "arxiv_id": "2608.30589",
  "arxiv_url": "https://arxiv.org/abs/2608.30589",
  "summary_what_math_gives_to_ml": "The paper provides a constructive way to turn scalar tensor-product wavelets into vector-valued bases whose basis elements mix signal components rather than processing channels independently. The transferable asset is a fixed, perfectly reconstructing multiscale analysis bank with compact support, vanishing moments, regularity, and structured cross-channel coupling. A practical neural-network use is a wavelet front end or downsampling block in which the cross-paired vector wavelets replace separate depthwise wavelet transforms, followed by lightweight learned channel mixing while retaining an exact inverse and a Parseval-style energy diagnostic.",
  "title": "Vector-Valued Wavelet Bases as Hilbert $\\mathbb{M}_m(\\mathbb{R})$-Module Bases: A Construction from Scalar Wavelets",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 4,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "memory"
 ],
 "title": "Cross-Channel Vector Wavelet Front End",
 "url": "https://synthcore.org/idea/2898/cross-channel-vector-wavelet-front-end",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}