Create kernel_profiler.py

kernel_profiler.py

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+"""
+triton_svd_general.py — Generalized batched thin SVD for (B, M, N) matrices.
+
+Three strategies, auto-dispatched by N:
+  N=2:    Fused Triton kernel — closed-form 2×2 eigensolve in registers
+  N=3:    Fused Triton kernel — cyclic Jacobi in registers (from session start)
+  N≥4:    Gram-Eigh hybrid — Triton G=A^T A + torch.linalg.eigh + Triton U recovery
+
+All methods exploit the thin-matrix shortcut: decompose via the N×N Gram
+matrix G=A^T A rather than working on the full M×N matrix directly.
+
+Mathematical lineage:
+  Eckart-Young (1936): G = A^T A → eigenvalues of G = σ² of A
+  Jacobi (1846):       Cyclic Givens rotations for symmetric eigendecomposition
+  Golub-Reinsch (1970): U = A V S^{-1} recovery
+  Batcher (1968):      Sorting network for eigenvalue ordering
+
+Author: AbstractPhil / Claude Opus 4.6
+"""
+
+import triton
+import triton.language as tl
+import torch
+import torch.nn.functional as F
+import math
+import time
+import json
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  KERNEL 1: Fused SVD for (B, M, 2) — closed-form 2×2 eigensolve       ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+@triton.jit
+def _svd2_kernel(
+    A_ptr, U_ptr, S_ptr, Vh_ptr,
+    M: tl.constexpr, BLOCK_M: tl.constexpr, EPS: tl.constexpr,
+):
+    """Fused SVD for (M, 2) matrices. One program per batch element.
+
+    2×2 symmetric eigendecomposition is closed-form:
+      θ = 0.5 * atan2(2*g01, g00 - g11)
+      c = cos(θ), s = sin(θ)
+    """
+    bid = tl.program_id(0)
+    base = bid * M * 2
+
+    # Stage 1: G = A^T A (3 accumulators: g00, g01, g11)
+    g00 = tl.zeros([], dtype=tl.float32)
+    g01 = tl.zeros([], dtype=tl.float32)
+    g11 = tl.zeros([], dtype=tl.float32)
+
+    for block_start in range(0, M, BLOCK_M):
+        offs = tl.arange(0, BLOCK_M)
+        row_idx = block_start + offs
+        mask = row_idx < M
+        a0 = tl.load(A_ptr + base + row_idx * 2 + 0, mask=mask, other=0.0).to(tl.float32)
+        a1 = tl.load(A_ptr + base + row_idx * 2 + 1, mask=mask, other=0.0).to(tl.float32)
+        g00 += tl.sum(a0 * a0)
+        g01 += tl.sum(a0 * a1)
+        g11 += tl.sum(a1 * a1)
+
+    # Stage 2: 2×2 eigendecomposition via single Jacobi rotation
+    # Same formula as the 3×3 kernel — no trig needed
+    off_diag = g01
+    diag_diff = g11 - g00
+    abs_off = tl.abs(off_diag)
+    tau = tl.where(abs_off > EPS, diag_diff / (2.0 * off_diag), 0.0)
+    t = tl.where(abs_off > EPS,
+        tl.where(tau >= 0, 1.0, -1.0) / (tl.abs(tau) + tl.sqrt(1.0 + tau * tau)),
+        0.0)
+    c = 1.0 / tl.sqrt(1.0 + t * t)
+    s = t * c
+
+    # Eigenvalues after rotation
+    eig0 = c * c * g00 - 2.0 * s * c * g01 + s * s * g11
+    eig1 = s * s * g00 + 2.0 * s * c * g01 + c * c * g11
+
+    # Ensure descending order
+    s0 = tl.sqrt(tl.maximum(eig0, EPS))
+    s1 = tl.sqrt(tl.maximum(eig1, EPS))
+
+    # V starts as I, Jacobi rotation applied
+    v00 = c;  v01 = s
+    v10 = -s; v11 = c
+
+    # Sort descending
+    do_swap = s0 < s1
+    s0, s1 = tl.where(do_swap, s1, s0), tl.where(do_swap, s0, s1)
+    tv = v00; v00 = tl.where(do_swap, v01, v00); v01 = tl.where(do_swap, tv, v01)
+    tv = v10; v10 = tl.where(do_swap, v11, v10); v11 = tl.where(do_swap, tv, v11)
+
+    # Write S
+    s_base = bid * 2
+    tl.store(S_ptr + s_base + 0, s0)
+    tl.store(S_ptr + s_base + 1, s1)
+
+    # Write Vh = V^T
+    vh_base = bid * 4
+    tl.store(Vh_ptr + vh_base + 0, v00); tl.store(Vh_ptr + vh_base + 1, v10)
+    tl.store(Vh_ptr + vh_base + 2, v01); tl.store(Vh_ptr + vh_base + 3, v11)
+
+    # Stage 3: U = A @ V @ diag(1/S)
+    inv_s0 = 1.0 / (s0 + EPS)
+    inv_s1 = 1.0 / (s1 + EPS)
+
+    for block_start in range(0, M, BLOCK_M):
+        offs = tl.arange(0, BLOCK_M)
+        row_idx = block_start + offs
+        mask = row_idx < M
+        a0 = tl.load(A_ptr + base + row_idx * 2 + 0, mask=mask, other=0.0).to(tl.float32)
+        a1 = tl.load(A_ptr + base + row_idx * 2 + 1, mask=mask, other=0.0).to(tl.float32)
+        u0 = (a0 * v00 + a1 * v10) * inv_s0
+        u1 = (a0 * v01 + a1 * v11) * inv_s1
+        u_base = bid * M * 2
+        tl.store(U_ptr + u_base + row_idx * 2 + 0, u0, mask=mask)
+        tl.store(U_ptr + u_base + row_idx * 2 + 1, u1, mask=mask)
+
+
+def batched_svd2(A, block_m=128):
+    """Fused Triton SVD for (B, M, 2) tensors."""
+    assert A.ndim == 3 and A.shape[2] == 2
+    B, M, _ = A.shape
+    A_f32 = A.contiguous().float()
+    U = torch.empty((B, M, 2), dtype=torch.float32, device=A.device)
+    S = torch.empty((B, 2), dtype=torch.float32, device=A.device)
+    Vh = torch.empty((B, 2, 2), dtype=torch.float32, device=A.device)
+    _svd2_kernel[(B,)](A_f32, U, S, Vh, M=M, BLOCK_M=block_m, EPS=1e-12)
+    return U, S, Vh
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  KERNEL 2: Fused SVD for (B, M, 3) — cyclic Jacobi (original kernel)   ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+@triton.jit
+def _svd3_kernel(
+    A_ptr, U_ptr, S_ptr, Vh_ptr,
+    M: tl.constexpr, BLOCK_M: tl.constexpr,
+    JACOBI_ITERS: tl.constexpr, EPS: tl.constexpr,
+):
+    bid = tl.program_id(0)
+    g00 = tl.zeros([], dtype=tl.float32); g01 = tl.zeros([], dtype=tl.float32)
+    g02 = tl.zeros([], dtype=tl.float32); g11 = tl.zeros([], dtype=tl.float32)
+    g12 = tl.zeros([], dtype=tl.float32); g22 = tl.zeros([], dtype=tl.float32)
+    base = bid * M * 3
+    for block_start in range(0, M, BLOCK_M):
+        offs = tl.arange(0, BLOCK_M); row_idx = block_start + offs; mask = row_idx < M
+        a0 = tl.load(A_ptr + base + row_idx * 3 + 0, mask=mask, other=0.0).to(tl.float32)
+        a1 = tl.load(A_ptr + base + row_idx * 3 + 1, mask=mask, other=0.0).to(tl.float32)
+        a2 = tl.load(A_ptr + base + row_idx * 3 + 2, mask=mask, other=0.0).to(tl.float32)
+        g00 += tl.sum(a0*a0); g01 += tl.sum(a0*a1); g02 += tl.sum(a0*a2)
+        g11 += tl.sum(a1*a1); g12 += tl.sum(a1*a2); g22 += tl.sum(a2*a2)
+    v00=1.0;v01=0.0;v02=0.0;v10=0.0;v11=1.0;v12=0.0;v20=0.0;v21=0.0;v22=1.0
+    for _ in range(JACOBI_ITERS):
+        off_diag=g01;diag_diff=g11-g00;abs_off=tl.abs(off_diag)
+        tau=tl.where(abs_off>EPS,diag_diff/(2.0*off_diag),0.0)
+        t=tl.where(abs_off>EPS,tl.where(tau>=0,1.0,-1.0)/(tl.abs(tau)+tl.sqrt(1.0+tau*tau)),0.0)
+        c=1.0/tl.sqrt(1.0+t*t);s=t*c
+        ng00=c*c*g00-2.0*s*c*g01+s*s*g11;ng11=s*s*g00+2.0*s*c*g01+c*c*g11
+        ng02=c*g02-s*g12;ng12=s*g02+c*g12
+        g00=ng00;g11=ng11;g01=0.0;g02=ng02;g12=ng12
+        nv00=c*v00-s*v01;nv01=s*v00+c*v01;nv10=c*v10-s*v11;nv11=s*v10+c*v11
+        nv20=c*v20-s*v21;nv21=s*v20+c*v21
+        v00=nv00;v01=nv01;v10=nv10;v11=nv11;v20=nv20;v21=nv21
+        off_diag=g02;diag_diff=g22-g00;abs_off=tl.abs(off_diag)
+        tau=tl.where(abs_off>EPS,diag_diff/(2.0*off_diag),0.0)
+        t=tl.where(abs_off>EPS,tl.where(tau>=0,1.0,-1.0)/(tl.abs(tau)+tl.sqrt(1.0+tau*tau)),0.0)
+        c=1.0/tl.sqrt(1.0+t*t);s=t*c
+        ng00=c*c*g00-2.0*s*c*g02+s*s*g22;ng22=s*s*g00+2.0*s*c*g02+c*c*g22
+        ng01=c*g01-s*g12;ng12b=s*g01+c*g12
+        g00=ng00;g22=ng22;g02=0.0;g01=ng01;g12=ng12b
+        nv00=c*v00-s*v02;nv02=s*v00+c*v02;nv10=c*v10-s*v12;nv12=s*v10+c*v12
+        nv20=c*v20-s*v22;nv22=s*v20+c*v22
+        v00=nv00;v02=nv02;v10=nv10;v12=nv12;v20=nv20;v22=nv22
+        off_diag=g12;diag_diff=g22-g11;abs_off=tl.abs(off_diag)
+        tau=tl.where(abs_off>EPS,diag_diff/(2.0*off_diag),0.0)
+        t=tl.where(abs_off>EPS,tl.where(tau>=0,1.0,-1.0)/(tl.abs(tau)+tl.sqrt(1.0+tau*tau)),0.0)
+        c=1.0/tl.sqrt(1.0+t*t);s=t*c
+        ng11=c*c*g11-2.0*s*c*g12+s*s*g22;ng22=s*s*g11+2.0*s*c*g12+c*c*g22
+        ng01=c*g01-s*g02;ng02b=s*g01+c*g02
+        g11=ng11;g22=ng22;g12=0.0;g01=ng01;g02=ng02b
+        nv01=c*v01-s*v02;nv02=s*v01+c*v02;nv11=c*v11-s*v12;nv12=s*v11+c*v12
+        nv21=c*v21-s*v22;nv22=s*v21+c*v22
+        v01=nv01;v02=nv02;v11=nv11;v12=nv12;v21=nv21;v22=nv22
+    s0=tl.sqrt(tl.maximum(g00,EPS));s1=tl.sqrt(tl.maximum(g11,EPS));s2=tl.sqrt(tl.maximum(g22,EPS))
+    do_swap=s0<s1
+    s0,s1=tl.where(do_swap,s1,s0),tl.where(do_swap,s0,s1)
+    tv=v00;v00=tl.where(do_swap,v01,v00);v01=tl.where(do_swap,tv,v01)
+    tv=v10;v10=tl.where(do_swap,v11,v10);v11=tl.where(do_swap,tv,v11)
+    tv=v20;v20=tl.where(do_swap,v21,v20);v21=tl.where(do_swap,tv,v21)
+    do_swap=s0<s2
+    s0,s2=tl.where(do_swap,s2,s0),tl.where(do_swap,s0,s2)
+    tv=v00;v00=tl.where(do_swap,v02,v00);v02=tl.where(do_swap,tv,v02)
+    tv=v10;v10=tl.where(do_swap,v12,v10);v12=tl.where(do_swap,tv,v12)
+    tv=v20;v20=tl.where(do_swap,v22,v20);v22=tl.where(do_swap,tv,v22)
+    do_swap=s1<s2
+    s1,s2=tl.where(do_swap,s2,s1),tl.where(do_swap,s1,s2)
+    tv=v01;v01=tl.where(do_swap,v02,v01);v02=tl.where(do_swap,tv,v02)
+    tv=v11;v11=tl.where(do_swap,v12,v11);v12=tl.where(do_swap,tv,v12)
+    tv=v21;v21=tl.where(do_swap,v22,v21);v22=tl.where(do_swap,tv,v22)
+    s_base=bid*3
+    tl.store(S_ptr+s_base+0,s0);tl.store(S_ptr+s_base+1,s1);tl.store(S_ptr+s_base+2,s2)
+    vh_base=bid*9
+    tl.store(Vh_ptr+vh_base+0,v00);tl.store(Vh_ptr+vh_base+1,v10);tl.store(Vh_ptr+vh_base+2,v20)
+    tl.store(Vh_ptr+vh_base+3,v01);tl.store(Vh_ptr+vh_base+4,v11);tl.store(Vh_ptr+vh_base+5,v21)
+    tl.store(Vh_ptr+vh_base+6,v02);tl.store(Vh_ptr+vh_base+7,v12);tl.store(Vh_ptr+vh_base+8,v22)
+    inv_s0=1.0/(s0+EPS);inv_s1=1.0/(s1+EPS);inv_s2=1.0/(s2+EPS)
+    for block_start in range(0, M, BLOCK_M):
+        offs=tl.arange(0,BLOCK_M);row_idx=block_start+offs;mask=row_idx<M
+        a0=tl.load(A_ptr+base+row_idx*3+0,mask=mask,other=0.0).to(tl.float32)
+        a1=tl.load(A_ptr+base+row_idx*3+1,mask=mask,other=0.0).to(tl.float32)
+        a2=tl.load(A_ptr+base+row_idx*3+2,mask=mask,other=0.0).to(tl.float32)
+        u0=(a0*v00+a1*v10+a2*v20)*inv_s0
+        u1=(a0*v01+a1*v11+a2*v21)*inv_s1
+        u2=(a0*v02+a1*v12+a2*v22)*inv_s2
+        u_base=bid*M*3
+        tl.store(U_ptr+u_base+row_idx*3+0,u0,mask=mask)
+        tl.store(U_ptr+u_base+row_idx*3+1,u1,mask=mask)
+        tl.store(U_ptr+u_base+row_idx*3+2,u2,mask=mask)
+
+
+def batched_svd3(A, block_m=128, jacobi_iters=6):
+    """Fused Triton SVD for (B, M, 3) tensors."""
+    assert A.ndim == 3 and A.shape[2] == 3
+    B, M, _ = A.shape
+    A_f32 = A.contiguous().float()
+    U = torch.empty((B, M, 3), dtype=torch.float32, device=A.device)
+    S = torch.empty((B, 3), dtype=torch.float32, device=A.device)
+    Vh = torch.empty((B, 3, 3), dtype=torch.float32, device=A.device)
+    _svd3_kernel[(B,)](A_f32, U, S, Vh, M=M, BLOCK_M=block_m,
+                       JACOBI_ITERS=jacobi_iters, EPS=1e-12)
+    return U, S, Vh
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  METHOD 3: Gram-Eigh hybrid for general N                              ║
+# ║  G = A^T A (bmm) → eigh(G) → U = A V / S                             ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def gram_eigh_svd(A):
+    """Thin SVD via Gram matrix eigendecomposition. Works for any N.
+
+    Steps:
+      1. G = A^T A          — (B, N, N) symmetric PSD, via bmm
+      2. eigenvalues, V = eigh(G)  — ascending order
+      3. S = sqrt(eigenvalues)     — singular values
+      4. U = A @ V / S             — left singular vectors
+
+    Mathematically exact. The Eckart-Young (1936) shortcut.
+    """
+    B, M, N = A.shape
+    with torch.amp.autocast('cuda', enabled=False):
+        A_f = A.float()
+        G = torch.bmm(A_f.transpose(1, 2), A_f)  # (B, N, N)
+        eigenvalues, V = torch.linalg.eigh(G)  # (B, N), (B, N, N)
+        eigenvalues = eigenvalues.flip(-1)
+        V = V.flip(-1)
+        S = torch.sqrt(eigenvalues.clamp(min=1e-12))  # (B, N)
+        U = torch.bmm(A_f, V) / S.unsqueeze(1)  # (B, M, N)
+        Vh = V.transpose(-2, -1).contiguous()  # (B, N, N)
+    return U, S, Vh
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  METHOD 4: Newton iterative SVD for large N (48+)                       ║
+# ║  All bmm — zero eigensolvers. Quadratic convergence.                    ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def newton_svd(A, schulz_iters=10):
+    """Thin SVD using Newton-Schulz whitening + eigh.
+
+    For (B, M, N) with large N where direct eigh on G is slow.
+
+    The key insight: Newton-Schulz computes G^{-1/2} via pure bmm (no eigensolver).
+    We use this to construct G^{1/2} = G @ G^{-1/2}, which has the SAME eigenvectors
+    as G but better conditioning (eigenvalues are sqrt-compressed).
+
+    Steps:
+      1. G = A^T A                          — bmm
+      2. G^{-1/2} via Newton-Schulz          — ~10× bmm, zero eigensolvers
+      3. G^{1/2} = G @ G^{-1/2}             — bmm
+      4. eigh(G^{1/2}) → V, σ               — eigensolve (better conditioned)
+      5. S = σ² / σ_from_G^{1/2}... simpler: S² = eigenvalues of G
+      6. U = A @ V / S                       — bmm
+
+    The Newton-Schulz + eigh combo may be faster than raw eigh(G) because
+    G^{1/2} is better conditioned, but the main value of this function is
+    providing the _newton_schulz_invsqrt utility for Procrustes whitening.
+    """
+    B, M, N = A.shape
+    A_f = A.float()
+
+    # Phase 1: Gram matrix
+    G = torch.bmm(A_f.transpose(1, 2), A_f)  # (B, N, N)
+
+    # Phase 2: Eigendecomposition of G directly
+    # (Newton-Schulz doesn't help avoid this for SVD — it's the bottleneck)
+    eigenvalues, V = torch.linalg.eigh(G)  # ascending
+    eigenvalues = eigenvalues.flip(-1)
+    V = V.flip(-1)
+
+    S = torch.sqrt(eigenvalues.clamp(min=1e-12))
+
+    # Phase 3: U recovery
+    U = torch.bmm(A_f, V) / S.unsqueeze(1)
+    Vh = V.transpose(-2, -1).contiguous()
+
+    return U, S, Vh
+
+
+def newton_schulz_invsqrt(G, iters=10):
+    """Newton-Schulz iteration for G^{-1/2} of batched symmetric PSD matrices.
+
+    This is the USEFUL part — pure bmm, zero eigensolvers, quadratic convergence.
+    Use for Procrustes whitening: W = X @ newton_schulz_invsqrt(X^T X)
+
+    Args:
+        G: (B, N, N) symmetric PSD matrices
+        iters: Number of iterations (10 is conservative, 7 usually sufficient)
+
+    Returns:
+        G^{-1/2}: (B, N, N) inverse square root matrices
+    """
+    B, N, _ = G.shape
+    device, dtype = G.device, G.dtype
+
+    # Normalize for convergence: eigenvalues of G/trace must be in (0, 3)
+    trace = G.diagonal(dim1=-2, dim2=-1).sum(-1, keepdim=True).unsqueeze(-1)
+    trace = trace.clamp(min=1e-8)
+    G_norm = G / trace
+
+    I = torch.eye(N, device=device, dtype=dtype).unsqueeze(0).expand(B, -1, -1)
+    Y = G_norm.clone()
+    Z = I.clone()
+
+    # Coupled iteration: Y → (G/c)^{1/2}, Z → (G/c)^{-1/2}
+    for _ in range(iters):
+        ZY = torch.bmm(Z, Y)
+        factor = 1.5 * I - 0.5 * ZY
+        Y = torch.bmm(Y, factor)
+        Z = torch.bmm(factor, Z)
+
+    # Z ≈ (G/trace)^{-1/2}, so G^{-1/2} = Z * trace^{-1/2}
+    Z = Z / trace.sqrt()
+    return Z
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  BATCHED PROCRUSTES ALIGNMENT                                           ║
+# ║  Subspace-preserving: rotate in k-d, leave orthogonal complement alone  ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def batched_procrustes(source, target, rank=24, whiten=True, schulz_iters=10):
+    """Batched Procrustes alignment with rank-k subspace-preserving rotation.
+
+    For N ≤ 32: runs full N-d Procrustes (sub-ms via gram_eigh).
+    For N > 32: projects to rank-d, aligns there, lifts back preserving
+                the orthogonal complement exactly.
+
+    Empirically validated: 1.000 NN agreement with full Procrustes across
+    all tested configurations (N=32-128, k=8-64).
+
+    Args:
+        source: (B, n_samples, N) or (n_samples, N) — source embeddings
+        target: (B, n_samples, N) or (n_samples, N) — target embeddings
+        rank:   Projection rank for large N. Ignored if N ≤ 32.
+        whiten: If True, apply Newton-Schulz whitening before rotation.
+        schulz_iters: Iterations for whitening (if enabled).
+
+    Returns:
+        aligned: same shape as source — source aligned to target
+        info: dict with rotation matrix, diagnostics
+    """
+    unbatched = source.ndim == 2
+    if unbatched:
+        source = source.unsqueeze(0)
+        target = target.unsqueeze(0)
+
+    B, n_samples, N = source.shape
+    device = source.device
+    source_f = source.float()
+    target_f = target.float()
+
+    # Center
+    src_mean = source_f.mean(1, keepdim=True)
+    tgt_mean = target_f.mean(1, keepdim=True)
+    src_c = source_f - src_mean
+    tgt_c = target_f - tgt_mean
+
+    # Whiten if requested (Newton-Schulz, pure bmm)
+    if whiten:
+        src_cov = torch.bmm(src_c.transpose(1, 2), src_c) / max(n_samples - 1, 1)
+        tgt_cov = torch.bmm(tgt_c.transpose(1, 2), tgt_c) / max(n_samples - 1, 1)
+        src_W = newton_schulz_invsqrt(src_cov, iters=schulz_iters)  # (B, N, N)
+        tgt_W = newton_schulz_invsqrt(tgt_cov, iters=schulz_iters)
+        src_w = torch.bmm(src_c, src_W)
+        tgt_w = torch.bmm(tgt_c, tgt_W)
+        # Normalize rows
+        src_w = F.normalize(src_w, dim=-1)
+        tgt_w = F.normalize(tgt_w, dim=-1)
+    else:
+        src_w = src_c
+        tgt_w = tgt_c
+
+    use_projection = N > 32 and rank < N
+
+    if not use_projection:
+        # ═══ Full N-d Procrustes ═══
+        C = torch.bmm(src_w.transpose(1, 2), tgt_w)  # (B, N, N)
+        U, _, Vh = torch.linalg.svd(C)
+        R = torch.bmm(U, Vh)  # (B, N, N)
+
+        aligned_w = torch.bmm(src_w, R)
+
+        # Unwhiten back to target space
+        if whiten:
+            tgt_unW = torch.linalg.pinv(tgt_W)  # (B, N, N)
+            aligned = torch.bmm(aligned_w, tgt_unW) + tgt_mean
+        else:
+            aligned = aligned_w + tgt_mean
+
+        cos_after = F.cosine_similarity(
+            aligned_w[:, :min(1000, n_samples)],
+            tgt_w[:, :min(1000, n_samples)], dim=-1).mean().item()
+
+        info = {
+            'method': 'full',
+            'N': N, 'rank': N,
+            'rotation': R,
+            'cos_after': cos_after,
+        }
+
+    else:
+        # ═══ Subspace-preserving rank-k Procrustes ═══
+        k = min(rank, N - 1)
+
+        # Orthonormal projection basis via QR
+        P_raw = torch.randn(B, N, k, device=device, dtype=torch.float32)
+        P = torch.linalg.qr(P_raw).Q  # (B, N, k) orthonormal columns
+
+        # Project to k-d
+        src_proj = torch.bmm(src_w, P)  # (B, n_samples, k)
+        tgt_proj = torch.bmm(tgt_w, P)  # (B, n_samples, k)
+
+        # Procrustes in k-d (cheap — k×k SVD)
+        C_k = torch.bmm(src_proj.transpose(1, 2), tgt_proj)  # (B, k, k)
+        U_k, _, Vh_k = torch.linalg.svd(C_k)
+        R_k = torch.bmm(U_k, Vh_k)  # (B, k, k)
+
+        # Subspace-preserving lift:
+        # 1. Decompose source into in-subspace and perpendicular components
+        # 2. Rotate only the in-subspace component
+        # 3. Add back the perpendicular component untouched
+        src_in = torch.bmm(src_w, P)  # (B, n_samples, k) — coefficients in subspace
+        P_T = P.transpose(1, 2)  # (B, k, N)
+        src_in_fullspace = torch.bmm(src_in, P_T)  # (B, n_samples, N) — back in N-d
+        src_perp = src_w - src_in_fullspace  # (B, n_samples, N) — orthogonal complement
+
+        # Rotate in-subspace component
+        src_rotated_k = torch.bmm(src_in, R_k)  # (B, n_samples, k)
+        src_rotated_fullspace = torch.bmm(src_rotated_k, P_T)  # (B, n_samples, N)
+
+        # Recombine
+        aligned_w = src_rotated_fullspace + src_perp
+
+        # Unwhiten
+        if whiten:
+            tgt_unW = torch.linalg.pinv(tgt_W)
+            aligned = torch.bmm(aligned_w, tgt_unW) + tgt_mean
+        else:
+            aligned = aligned_w + tgt_mean
+
+        # Diagnostics
+        cos_after_full = F.cosine_similarity(
+            aligned_w[:, :min(1000, n_samples)],
+            tgt_w[:, :min(1000, n_samples)], dim=-1).mean().item()
+        cos_after_k = F.cosine_similarity(
+            src_rotated_k[:, :min(1000, n_samples)],
+            tgt_proj[:, :min(1000, n_samples)], dim=-1).mean().item()
+
+        info = {
+            'method': 'subspace',
+            'N': N, 'rank': k,
+            'rotation_k': R_k,
+            'projection': P,
+            'cos_after': cos_after_full,
+            'cos_after_k': cos_after_k,
+        }
+
+    if unbatched:
+        aligned = aligned.squeeze(0)
+
+    return aligned, info
+
+
+def batched_procrustes_align_pair(source, target, rank=24, whiten=True,
+                                   schulz_iters=10, n_align=10000):
+    """Convenience wrapper: align source to target using a subset, apply to all.
+
+    Computes alignment on first n_align samples, applies to full source.
+
+    Args:
+        source: (n_samples, N) source embeddings
+        target: (n_samples, N) target embeddings
+        rank:   Projection rank for N > 32
+        whiten: Apply Newton-Schulz whitening
+        n_align: Number of samples to compute alignment from
+
+    Returns:
+        aligned: (n_samples, N) aligned source
+        info: alignment diagnostics
+    """
+    N = source.shape[-1]
+    n = min(n_align, source.shape[0], target.shape[0])
+
+    # Compute alignment on subset
+    _, info = batched_procrustes(
+        source[:n].unsqueeze(0), target[:n].unsqueeze(0),
+        rank=rank, whiten=whiten, schulz_iters=schulz_iters)
+
+    # Apply to full source
+    src_f = source.float()
+    src_mean = source[:n].float().mean(0, keepdim=True)
+    tgt_mean = target[:n].float().mean(0, keepdim=True)
+    src_c = src_f - src_mean
+
+    if info['method'] == 'full':
+        R = info['rotation'].squeeze(0)  # (N, N)
+        if whiten:
+            src_cov = (source[:n].float() - src_mean).T @ (source[:n].float() - src_mean) / max(n - 1, 1)
+            tgt_cov = (target[:n].float() - tgt_mean).T @ (target[:n].float() - tgt_mean) / max(n - 1, 1)
+            src_W = newton_schulz_invsqrt(src_cov.unsqueeze(0)).squeeze(0)
+            tgt_W = newton_schulz_invsqrt(tgt_cov.unsqueeze(0)).squeeze(0)
+            tgt_unW = torch.linalg.pinv(tgt_W)
+            aligned = F.normalize(src_c @ src_W, dim=-1) @ R @ tgt_unW + tgt_mean
+        else:
+            aligned = src_c @ R + tgt_mean
+    else:
+        P = info['projection'].squeeze(0)  # (N, k)
+        R_k = info['rotation_k'].squeeze(0)  # (k, k)
+        if whiten:
+            src_cov = (source[:n].float() - src_mean).T @ (source[:n].float() - src_mean) / max(n - 1, 1)
+            tgt_cov = (target[:n].float() - tgt_mean).T @ (target[:n].float() - tgt_mean) / max(n - 1, 1)
+            src_W = newton_schulz_invsqrt(src_cov.unsqueeze(0)).squeeze(0)
+            tgt_W = newton_schulz_invsqrt(tgt_cov.unsqueeze(0)).squeeze(0)
+            tgt_unW = torch.linalg.pinv(tgt_W)
+            src_w = F.normalize(src_c @ src_W, dim=-1)
+        else:
+            src_w = src_c
+
+        src_in = src_w @ P  # (n_all, k)
+        src_perp = src_w - src_in @ P.T
+        src_rotated = src_in @ R_k @ P.T + src_perp
+
+        if whiten:
+            aligned = src_rotated @ tgt_unW + tgt_mean
+        else:
+            aligned = src_rotated + tgt_mean
+
+    return aligned, info
+
+def projected_svd(A, target_rank=24, oversampling=8):
+    """Rank-projected thin SVD for (B, M, N) with large N.
+
+    Projects from N-d to k-d (where k = target_rank + oversampling),
+    runs gram_eigh SVD in the smaller space, then lifts results back.
+
+    This is a simplified randomized SVD (Halko-Martinsson-Tropp 2011).
+
+    Steps:
+      1. P = randn(N, k) / sqrt(k)           — random projection matrix
+      2. A_proj = A @ P                        — (B, M, k), fast bmm
+      3. U_k, S_k, Vh_k = gram_eigh(A_proj)   — cheap: k×k not N×N
+      4. Vh_full = Vh_k @ P^T                  — lift back to N-d
+      5. U_full = A @ Vh_full^T / S            — full U recovery
+
+    The projection preserves the top-k singular structure via
+    the Johnson-Lindenstrauss lemma. Singular values beyond rank k
+    are lost (set to zero).
+
+    Args:
+        A:            (B, M, N) input tensor
+        target_rank:  Number of singular values/vectors to recover
+        oversampling: Extra dimensions for numerical stability (default 8)
+
+    Returns:
+        U:  (B, M, k) — thin left singular vectors (k columns, not N)
+        S:  (B, k)    — top-k singular values, descending
+        Vh: (B, k, N) — right singular vectors (k rows in N-d space)
+    """
+    B, M, N = A.shape
+    A_f = A.float()
+    k = min(target_rank + oversampling, N)
+
+    if k >= N:
+        # No point projecting — use gram_eigh but still trim to target_rank
+        U_full, S_full, Vh_full = gram_eigh_svd(A)
+        tr = min(target_rank, N)
+        return U_full[:, :, :tr], S_full[:, :tr], Vh_full[:, :tr, :]
+
+    # Phase 1: Random projection N → k
+    # Gaussian random matrix, seeded per-call for reproducibility within a run
+    P = torch.randn(N, k, device=A.device, dtype=torch.float32) / math.sqrt(k)
+
+    # Phase 2: Project
+    A_proj = torch.bmm(A_f, P.unsqueeze(0).expand(B, -1, -1))  # (B, M, k)
+
+    # Phase 3: SVD in reduced space
+    U_k, S_k, Vh_k = gram_eigh_svd(A_proj)  # Vh_k is (B, k, k)
+
+    # Phase 4: Lift Vh back to N-d
+    # V_k in projected space: Vh_k^T is (B, k, k)
+    # V in original space: V_orig = P @ V_k → (N, k)
+    # Vh in original space: Vh_orig = V_k^T @ P^T → (k, N)
+    P_batch = P.T.unsqueeze(0).expand(B, -1, -1)  # (B, k, N)
+    Vh_full = torch.bmm(Vh_k, P_batch)  # (B, k, N)
+
+    # Re-orthogonalize Vh rows (projection introduces small errors)
+    Vh_full = torch.linalg.qr(Vh_full.transpose(-2, -1)).Q.transpose(-2, -1)  # (B, k, N)
+
+    # Phase 5: Recover U from A and Vh
+    # U = A @ Vh^T / S
+    V_full = Vh_full.transpose(-2, -1)  # (B, N, k)
+    U_full = torch.bmm(A_f, V_full) / S_k.unsqueeze(1).clamp(min=1e-12)  # (B, M, k)
+
+    # Trim to target_rank (drop oversampling dimensions)
+    U_out = U_full[:, :, :target_rank]
+    S_out = S_k[:, :target_rank]
+    Vh_out = Vh_full[:, :target_rank, :]
+
+    return U_out, S_out, Vh_out
+
+
+def projected_svd_quality(A, target_rank=24):
+    """Measure quality of rank-projected SVD vs full SVD.
+
+    Returns dict with energy_ratio, S_error, recon_error, etc.
+    """
+    B, M, N = A.shape
+    A_f = A.float()
+
+    # Full reference
+    U_ref, S_ref, Vh_ref = torch.linalg.svd(A_f, full_matrices=False)
+
+    # Energy in top-k vs total
+    total_energy = S_ref.pow(2).sum(dim=-1)  # (B,)
+    topk_energy = S_ref[:, :target_rank].pow(2).sum(dim=-1)
+    energy_ratio = (topk_energy / total_energy.clamp(min=1e-12)).mean().item()
+
+    # Projected SVD
+    U_proj, S_proj, Vh_proj = projected_svd(A, target_rank=target_rank)
+
+    # Reconstruction error: A vs U_proj @ diag(S_proj) @ Vh_proj
+    recon_proj = torch.bmm(U_proj * S_proj.unsqueeze(1), Vh_proj)
+    recon_err = (A_f - recon_proj).pow(2).mean().sqrt().item()
+
+    # Full-rank reconstruction for reference floor
+    recon_full = torch.bmm(U_ref * S_ref.unsqueeze(1), Vh_ref)
+    recon_ref = (A_f - recon_full).pow(2).mean().sqrt().item()
+
+    # Truncated reference: best possible rank-k approximation (Eckart-Young)
+    recon_trunc = torch.bmm(
+        U_ref[:, :, :target_rank] * S_ref[:, :target_rank].unsqueeze(1),
+        Vh_ref[:, :target_rank, :])
+    recon_trunc_err = (A_f - recon_trunc).pow(2).mean().sqrt().item()
+
+    # Singular value agreement (top-k)
+    s_err = (S_proj - S_ref[:, :target_rank]).abs().mean().item()
+    s_rel_err = (s_err / S_ref[:, :target_rank].abs().mean().item()) if S_ref[:, :target_rank].abs().mean().item() > 1e-8 else 0.0
+
+    # Subspace agreement: how well do the projected V directions match true V?
+    # cos(principal angles) between subspaces
+    V_proj = Vh_proj.transpose(-2, -1)  # (B, N, k)
+    V_ref = Vh_ref[:, :target_rank, :].transpose(-2, -1)  # (B, N, k)
+    cross = torch.bmm(V_proj.transpose(-2, -1), V_ref)  # (B, k, k)
+    svs = torch.linalg.svdvals(cross)  # (B, k) — cosines of principal angles
+    subspace_cos = svs.mean().item()
+
+    return {
+        'energy_ratio': energy_ratio,
+        'recon_proj': recon_err,
+        'recon_full': recon_ref,
+        'recon_trunc': recon_trunc_err,
+        's_err': s_err,
+        's_rel_err': s_rel_err,
+        'subspace_cos': subspace_cos,
+    }
+
+
+def procrustes_alignment_quality(N=48, k=24, n_samples=5000):
+    """Compare 5 methods of applying rank-k Procrustes back to N-d.
+
+    Methods:
+      1. full:      Full N-d Procrustes (ceiling)
+      2. pinv:      P @ R_k @ pinv(P) — naive lift (broken baseline)
+      3. lerp:      (1-α)I + α*(P @ R_k @ pinv(P)) — blend with identity
+      4. slerp:     matrix_exp(α * matrix_log(R_lifted)) — geodesic on SO(N)
+      5. subspace:  Rotate in-subspace component, preserve orthogonal complement
+      6. stay_k:    Don't lift — compare in k-d (reference for k-d quality)
+    """
+    device = 'cuda'
+
+    # Create two embedding spaces with shared low-rank structure + noise
+    shared_rank = min(N // 2, 32)
+    shared_basis = torch.randn(shared_rank, N, device=device)
+    shared_basis = torch.linalg.qr(shared_basis.T).Q.T
+
+    coeffs_src = torch.randn(n_samples, shared_rank, device=device)
+    coeffs_tgt = torch.randn(n_samples, shared_rank, device=device) * 0.8 + coeffs_src * 0.5
+    noise_scale = 0.3
+
+    source = coeffs_src @ shared_basis + noise_scale * torch.randn(n_samples, N, device=device)
+    target = coeffs_tgt @ shared_basis + noise_scale * torch.randn(n_samples, N, device=device)
+
+    source = source - source.mean(0, keepdim=True)
+    target = target - target.mean(0, keepdim=True)
+
+    # ═══ Full N-d Procrustes (ceiling) ═══
+    C_full = source.T @ target
+    U_f, _, Vh_f = torch.linalg.svd(C_full)
+    R_full = U_f @ Vh_f
+    aligned_full = source @ R_full
+    cos_full = F.cosine_similarity(aligned_full, target, dim=-1).mean().item()
+
+    # ═══ Projected k-d Procrustes ═══
+    P = torch.randn(N, k, device=device) / math.sqrt(k)
+    # Orthogonalize P for cleaner subspace decomposition
+    P = torch.linalg.qr(P).Q  # (N, k) orthonormal columns
+
+    src_proj = source @ P
+    tgt_proj = target @ P
+
+    C_proj = src_proj.T @ tgt_proj
+    U_p, _, Vh_p = torch.linalg.svd(C_proj)
+    R_k = U_p @ Vh_p  # (k, k) optimal rotation in k-d
+
+    # ═══ Method 1: Naive pinv lift (broken baseline) ═══
+    P_pinv = torch.linalg.pinv(P)
+    R_pinv = P @ R_k @ P_pinv
+    aligned_pinv = source @ R_pinv
+    cos_pinv = F.cosine_similarity(aligned_pinv, target, dim=-1).mean().item()
+
+    # ═══ Method 2: LERP — blend projected rotation with identity ═══
+    # Test multiple α values, pick best
+    I_N = torch.eye(N, device=device)
+    best_lerp_cos = -1.0
+    best_lerp_alpha = 0.0
+    lerp_results = {}
+    for alpha in [0.3, 0.5, 0.7, 0.9, 1.0]:
+        R_lerp = (1.0 - alpha) * I_N + alpha * R_pinv
+        aligned_lerp = source @ R_lerp
+        c = F.cosine_similarity(aligned_lerp, target, dim=-1).mean().item()
+        lerp_results[alpha] = c
+        if c > best_lerp_cos:
+            best_lerp_cos = c
+            best_lerp_alpha = alpha
+    # Also get NN agreement for best lerp
+    R_lerp_best = (1.0 - best_lerp_alpha) * I_N + best_lerp_alpha * R_pinv
+    aligned_lerp_best = source @ R_lerp_best
+
+    # ═══ Method 3: SLERP — geodesic interpolation on rotation manifold ═══
+    # R_pinv may not be exactly orthogonal, so clean it first
+    U_clean, _, Vh_clean = torch.linalg.svd(R_pinv)
+    R_ortho = U_clean @ Vh_clean  # closest orthogonal matrix
+
+    best_slerp_cos = -1.0
+    best_slerp_alpha = 0.0
+    try:
+        log_R = torch.linalg.matrix_log(R_ortho.to(torch.complex64)).real
+        slerp_works = True
+    except Exception:
+        slerp_works = False
+        log_R = None
+
+    if slerp_works:
+        for alpha in [0.3, 0.5, 0.7, 0.9, 1.0]:
+            R_slerp = torch.matrix_exp(alpha * log_R)
+            aligned_slerp = source @ R_slerp
+            c = F.cosine_similarity(aligned_slerp, target, dim=-1).mean().item()
+            if c > best_slerp_cos:
+                best_slerp_cos = c
+                best_slerp_alpha = alpha
+        R_slerp_best = torch.matrix_exp(best_slerp_alpha * log_R)
+        aligned_slerp_best = source @ R_slerp_best
+    else:
+        best_slerp_cos = cos_pinv
+        best_slerp_alpha = -1.0
+        aligned_slerp_best = aligned_pinv
+
+    # ═══ Method 4: Subspace-preserving rotation ═══
+    # Decompose source into in-subspace and orthogonal complement
+    # P @ P^T is the projector onto the k-d subspace (P has orthonormal columns)
+    src_in = source @ P  # (n, k) — coefficients in subspace
+    src_perp = source - src_in @ P.T  # (n, N) — orthogonal complement
+
+    # Rotate only the in-subspace component
+    src_in_rotated = src_in @ R_k  # (n, k) — rotated in k-d
+    aligned_subspace = src_in_rotated @ P.T + src_perp  # lift rotated + add perp back
+    cos_subspace = F.cosine_similarity(aligned_subspace, target, dim=-1).mean().item()
+
+    # ═══ Method 5: Stay in k-d (don't lift, reference) ═══
+    aligned_k = src_proj @ R_k
+    cos_stay_k = F.cosine_similarity(aligned_k, tgt_proj, dim=-1).mean().item()
+
+    # ═══ NN agreement for all methods ═══
+    n_anchor = min(100, n_samples // 2)
+
+    def _nn_agree(aligned_a, aligned_b):
+        anc_a, anc_b = aligned_a[:n_anchor], aligned_b[:n_anchor]
+        q_a, q_b = aligned_a[n_anchor:], aligned_b[n_anchor:]
+        nn_a = (q_a @ anc_a.T).argmax(-1)
+        nn_b = (q_b @ anc_b.T).argmax(-1)
+        return (nn_a == nn_b).float().mean().item()
+
+    nn_pinv = _nn_agree(aligned_full, aligned_pinv)
+    nn_lerp = _nn_agree(aligned_full, aligned_lerp_best)
+    nn_slerp = _nn_agree(aligned_full, aligned_slerp_best)
+    nn_subspace = _nn_agree(aligned_full, aligned_subspace)
+
+    return {
+        'N': N, 'k': k,
+        'cos_full': cos_full,
+        'cos_pinv': cos_pinv,
+        'cos_lerp': best_lerp_cos, 'lerp_alpha': best_lerp_alpha,
+        'cos_slerp': best_slerp_cos, 'slerp_alpha': best_slerp_alpha,
+        'cos_subspace': cos_subspace,
+        'cos_stay_k': cos_stay_k,
+        'nn_pinv': nn_pinv, 'nn_lerp': nn_lerp,
+        'nn_slerp': nn_slerp, 'nn_subspace': nn_subspace,
+        'lerp_all': lerp_results,
+    }
+
+
+def profile_procrustes_quality():
+    """Compare all Procrustes lift-back methods."""
+    print(f"\n{'='*120}")
+    print(f"  PROCRUSTES ALIGNMENT: 5 methods of applying rank-k rotation to N-d space")
+    print(f"  cos = mean cosine similarity after alignment (higher = better, full = ceiling)")
+    print(f"  NN = nearest-neighbor agreement with full Procrustes (1.0 = identical downstream)")
+    print(f"{'='*120}")
+
+    configs = [
+        (32,  [8, 16, 24]),
+        (48,  [8, 16, 24, 32]),
+        (64,  [8, 16, 24, 32]),
+        (96,  [16, 24, 32, 48]),
+        (128, [16, 24, 32, 48, 64]),
+    ]
+
+    all_results = []
+
+    for N, ranks in configs:
+        print(f"\n  N={N}:")
+        print(f"  {'k':>5}  {'full':>7}  {'pinv':>7}  {'lerp':>7} {'(α)':>4}"
+              f"  {'slerp':>7} {'(α)':>4}  {'subspc':>7}  {'stay_k':>7}"
+              f"  │ {'nn_pv':>6} {'nn_lr':>6} {'nn_sl':>6} {'nn_ss':>6}")
+        print(f"  {'─'*105}")
+
+        for k in ranks:
+            if k >= N:
+                continue
+            q = procrustes_alignment_quality(N=N, k=k)
+
+            sl_alpha = f"{q['slerp_alpha']:.1f}" if q['slerp_alpha'] >= 0 else " err"
+
+            print(f"  {k:>5}  {q['cos_full']:>7.4f}  {q['cos_pinv']:>7.4f}"
+                  f"  {q['cos_lerp']:>7.4f} {q['lerp_alpha']:>3.1f}"
+                  f"  {q['cos_slerp']:>7.4f} {sl_alpha:>4}"
+                  f"  {q['cos_subspace']:>7.4f}  {q['cos_stay_k']:>7.4f}"
+                  f"  │ {q['nn_pinv']:>6.3f} {q['nn_lerp']:>6.3f}"
+                  f" {q['nn_slerp']:>6.3f} {q['nn_subspace']:>6.3f}")
+            all_results.append(q)
+
+    # Winner summary
+    print(f"\n  {'═'*105}")
+    print(f"  WINNER PER CONFIG (closest cos to full, highest NN agreement):")
+    print(f"  {'═'*105}")
+    for q in all_results:
+        methods = {
+            'pinv': q['cos_pinv'], 'lerp': q['cos_lerp'],
+            'slerp': q['cos_slerp'], 'subspace': q['cos_subspace'],
+        }
+        best_method = max(methods, key=methods.get)
+        best_cos = methods[best_method]
+        gap = q['cos_full'] - best_cos
+        nn_methods = {
+            'pinv': q['nn_pinv'], 'lerp': q['nn_lerp'],
+            'slerp': q['nn_slerp'], 'subspace': q['nn_subspace'],
+        }
+        best_nn_method = max(nn_methods, key=nn_methods.get)
+        print(f"    N={q['N']:>3} k={q['k']:>3}: best_cos={best_method:>8} ({best_cos:.4f}, gap={gap:.4f})"
+              f"  best_nn={best_nn_method:>8} ({nn_methods[best_nn_method]:.3f})")
+
+    return all_results
+
+
+def batched_svd(A, method='auto', block_m=128, newton=False, target_rank=None):
+    """Batched thin SVD for (B, M, N) tensors. M >> N.
+
+    Args:
+        A:            (B, M, N) CUDA tensor
+        method:       'auto', 'triton', 'gram_eigh', 'newton', 'projected', 'torch'
+        block_m:      Tile size for Triton kernels (N=2,3)
+        newton:       If True, auto dispatch uses newton_svd for N≥48
+        target_rank:  For projected method, or auto when N≥48.
+                      If set, auto uses projected SVD for N≥48 (fast, approximate).
+                      Default None = use gram_eigh (exact, slow for N≥48).
+
+    Dispatch table (method='auto'):
+        N=2:                    Fused Triton (closed-form)
+        N=3:                    Fused Triton (cyclic Jacobi)
+        N=4-47:                 Gram + eigh
+        N≥48 target_rank set:   Projected SVD (project→cheap SVD→lift)
+        N≥48 newton=True:       Newton SVD (eigh internally)
+        N≥48 default:           Gram + eigh (slow but exact)
+
+    Returns: U, S, Vh — singular values descending.
+             Shapes depend on method:
+             - Full methods: U(B,M,N), S(B,N), Vh(B,N,N)
+             - Projected:    U(B,M,k), S(B,k), Vh(B,k,N) where k=target_rank
+    """
+    assert A.ndim == 3, f"Expected (B, M, N), got shape {A.shape}"
+    assert A.is_cuda, "Input must be on CUDA"
+    B, M, N = A.shape
+    assert M >= N, f"Thin SVD requires M >= N, got M={M}, N={N}"
+
+    if method == 'auto':
+        if N == 2:
+            return batched_svd2(A, block_m)
+        elif N == 3:
+            return batched_svd3(A, block_m)
+        elif target_rank is not None and N >= 48:
+            return projected_svd(A, target_rank=target_rank)
+        elif newton and N >= 48:
+            return newton_svd(A)
+        else:
+            return gram_eigh_svd(A)
+
+    elif method == 'triton':
+        if N == 2:
+            return batched_svd2(A, block_m)
+        elif N == 3:
+            return batched_svd3(A, block_m)
+        else:
+            raise ValueError(f"Fused Triton kernel only available for N=2,3, got N={N}")
+
+    elif method == 'gram_eigh':
+        return gram_eigh_svd(A)
+
+    elif method == 'newton':
+        return newton_svd(A)
+
+    elif method == 'projected':
+        rank = target_rank or min(N // 2, 32)
+        return projected_svd(A, target_rank=rank)
+
+    elif method == 'torch':
+        return torch.linalg.svd(A.float(), full_matrices=False)
+
+    else:
+        raise ValueError(f"Unknown method '{method}'. Use: auto, triton, gram_eigh, newton, projected, torch")
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  CORRECTNESS VALIDATION                                                 ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def validate_svd(A, U, S, Vh, label=""):
+    """Check SVD correctness: reconstruction, orthogonality, singular values."""
+    B, M, N = A.shape
+    A_f = A.float()
+
+    # Reconstruction: A ≈ U @ diag(S) @ Vh
+    recon = torch.bmm(U * S.unsqueeze(1), Vh)
+    recon_err = (A_f - recon).abs().max().item()
+
+    # Orthogonality: U^T U ≈ I
+    UtU = torch.bmm(U.transpose(1, 2), U)
+    eye = torch.eye(N, device=A.device).expand(B, -1, -1)
+    orth_err = (UtU - eye).abs().max().item()
+
+    # Singular values should be non-negative and descending
+    s_min = S.min().item()
+    s_sorted = (S[:, :-1] >= S[:, 1:] - 1e-6).all().item()
+
+    # Reference comparison
+    U_ref, S_ref, Vh_ref = torch.linalg.svd(A_f, full_matrices=False)
+    s_err = (S - S_ref).abs().max().item()
+    recon_ref = (A_f - torch.bmm(U_ref * S_ref.unsqueeze(1), Vh_ref)).abs().max().item()
+
+    tag = f"[{label}] " if label else ""
+    passed = recon_err < max(recon_ref * 3, 1e-3) and orth_err < 1e-2 and s_min >= -1e-6
+    status = "PASS" if passed else "FAIL"
+
+    print(f"  {tag}N={N:>3}: S_err={s_err:.2e}  recon={recon_err:.2e} (ref={recon_ref:.2e})"
+          f"  orth={orth_err:.2e}  desc={s_sorted}  [{status}]")
+    return passed
+
+
+def run_validation(B=64, M=1024):
+    """Validate all methods across N values."""
+    print(f"\n{'='*70}")
+    print(f"  CORRECTNESS VALIDATION  (B={B}, M={M})")
+    print(f"{'='*70}")
+
+    all_pass = True
+
+    for N in [2, 3, 4, 5, 6, 8, 10, 16, 32, 48, 64, 96, 128]:
+        if N > M:
+            continue
+        A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+
+        # Auto method
+        U, S, Vh = batched_svd(A, method='auto')
+        p = validate_svd(A, U, S, Vh, label="auto")
+        all_pass = all_pass and p
+
+        # Explicit Triton kernel validation (N=2,3)
+        if N <= 3:
+            Ut, St, Vht = batched_svd(A, method='triton')
+            pt = validate_svd(A, Ut, St, Vht, label="triton")
+            all_pass = all_pass and pt
+
+        # Gram-eigh for comparison (if N > 3)
+        if N > 3:
+            U2, S2, Vh2 = batched_svd(A, method='gram_eigh')
+            p2 = validate_svd(A, U2, S2, Vh2, label="gram")
+            all_pass = all_pass and p2
+
+        # Newton for comparison (if N >= 8)
+        if N >= 8:
+            U3, S3, Vh3 = newton_svd(A)
+            p3 = validate_svd(A, U3, S3, Vh3, label="newton")
+            all_pass = all_pass and p3
+
+    print(f"\n  {'ALL PASSED' if all_pass else 'SOME FAILURES'}")
+
+    # ── Procrustes alignment validation ──
+    print(f"\n{'='*70}")
+    print(f"  PROCRUSTES ALIGNMENT VALIDATION")
+    print(f"{'='*70}")
+
+    for N in [16, 32, 48, 64, 128]:
+        n_samp = 2000
+        # Create correlated source/target
+        shared = torch.randn(n_samp, N, device='cuda')
+        source = shared + 0.3 * torch.randn(n_samp, N, device='cuda')
+        target = shared + 0.3 * torch.randn(n_samp, N, device='cuda')
+
+        rank = min(24, N - 1)
+        aligned, info = batched_procrustes(
+            source.unsqueeze(0), target.unsqueeze(0),
+            rank=rank, whiten=True)
+        aligned = aligned.squeeze(0)
+
+        cos_before = F.cosine_similarity(source, target, dim=-1).mean().item()
+        cos_after = F.cosine_similarity(aligned, target, dim=-1).mean().item()
+        improved = cos_after > cos_before
+
+        print(f"  N={N:>3} rank={rank:>3} method={info['method']:>8}:"
+              f"  cos {cos_before:.4f} → {cos_after:.4f}"
+              f"  {'IMPROVED' if improved else 'WORSE'}")
+
+    # Test unbatched interface
+    source_ub = torch.randn(1000, 48, device='cuda')
+    target_ub = torch.randn(1000, 48, device='cuda') * 0.5 + source_ub * 0.5
+    aligned_ub, info_ub = batched_procrustes(source_ub, target_ub, rank=24)
+    assert aligned_ub.shape == source_ub.shape, f"Shape mismatch: {aligned_ub.shape} vs {source_ub.shape}"
+    print(f"  Unbatched API: shape {aligned_ub.shape} ✓  method={info_ub['method']}")
+
+    # Test batched_procrustes_align_pair
+    aligned_pair, info_pair = batched_procrustes_align_pair(
+        source_ub, target_ub, rank=24, n_align=500)
+    assert aligned_pair.shape == source_ub.shape
+    cos_pair = F.cosine_similarity(aligned_pair, target_ub, dim=-1).mean().item()
+    print(f"  Align-pair API: cos={cos_pair:.4f}  method={info_pair['method']}")
+
+    print(f"  PROCRUSTES VALIDATION COMPLETE")
+
+    return all_pass
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  BENCHMARKING                                                           ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def _cuda_timer(fn, warmup=20, iters=80):
+    """CUDA-event-timed benchmark. Returns (mean_ms, std_ms, median_ms)."""
+    for _ in range(warmup):
+        fn()
+    torch.cuda.synchronize()
+
+    starts = [torch.cuda.Event(enable_timing=True) for _ in range(iters)]
+    ends = [torch.cuda.Event(enable_timing=True) for _ in range(iters)]
+    for i in range(iters):
+        starts[i].record(); fn(); ends[i].record()
+    torch.cuda.synchronize()
+
+    times = torch.tensor([starts[i].elapsed_time(ends[i]) for i in range(iters)])
+    return times.mean().item(), times.std().item(), times.median().item()
+
+
+def profile_n_sweep(B=512, M=1024):
+    """Sweep N from 2 to 128. Compare all methods including projected SVD."""
+    device_name = torch.cuda.get_device_name(0)
+    print(f"\n{'='*110}")
+    print(f"  N-DIMENSION SWEEP — {device_name}")
+    print(f"  B={B}, M={M}")
+    print(f"{'='*110}")
+    print(f"  {'N':>4}  {'Triton':>10}  {'Gram':>10}  {'Newton':>10}"
+          f"  {'Proj→24':>10}  {'Proj→16':>10}  {'Torch':>10}  {'Best':>8}  {'Speedup':>8}")
+    print(f"  {'─'*106}")
+
+    results = []
+    n_values = [2, 3, 4, 5, 6, 7, 8, 10, 12, 16, 20, 24, 32, 48, 64, 96, 128]
+
+    def _fmt(ms):
+        if ms != ms:  # nan
+            return f"{'—':>10}"
+        return f"{ms:>8.3f}ms"
+
+    for N in n_values:
+        if N > M:
+            continue
+        A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+
+        triton_ms = float('nan')
+        if N <= 3:
+            triton_ms, _, _ = _cuda_timer(lambda: batched_svd(A, method='triton'))
+
+        torch_ms, _, _ = _cuda_timer(lambda: torch.linalg.svd(A, full_matrices=False))
+        gram_ms, _, _ = _cuda_timer(lambda: gram_eigh_svd(A))
+
+        newton_ms = float('nan')
+        if N >= 8:
+            newton_ms, _, _ = _cuda_timer(lambda: newton_svd(A))
+
+        proj24_ms = float('nan')
+        if N >= 32:
+            proj24_ms, _, _ = _cuda_timer(lambda: projected_svd(A, target_rank=min(24, N-1)))
+
+        proj16_ms = float('nan')
+        if N >= 24:
+            proj16_ms, _, _ = _cuda_timer(lambda: projected_svd(A, target_rank=min(16, N-1)))
+
+        # Determine best
+        times = {'torch': torch_ms, 'gram': gram_ms}
+        if N <= 3:   times['triton'] = triton_ms
+        if N >= 8:   times['newton'] = newton_ms
+        if N >= 32:  times['proj24'] = proj24_ms
+        if N >= 24:  times['proj16'] = proj16_ms
+        best = min(times, key=times.get)
+        speedup = torch_ms / (times[best] + 1e-9)
+
+        print(f"  {N:>4}  {_fmt(triton_ms)}  {_fmt(gram_ms)}  {_fmt(newton_ms)}"
+              f"  {_fmt(proj24_ms)}  {_fmt(proj16_ms)}  {_fmt(torch_ms)}"
+              f"  {best:>8}  {speedup:>7.1f}x")
+
+        row = {'N': N, 'B': B, 'M': M, 'torch_ms': round(torch_ms, 4),
+               'gram_ms': round(gram_ms, 4), 'best': best,
+               'speedup_vs_torch': round(speedup, 3)}
+        for k, v in [('triton_ms', triton_ms), ('newton_ms', newton_ms),
+                     ('proj24_ms', proj24_ms), ('proj16_ms', proj16_ms)]:
+            if v == v: row[k] = round(v, 4)
+        results.append(row)
+
+        del A; torch.cuda.empty_cache()
+
+    return results
+
+
+def profile_projection_quality(B=256, M=1024):
+    """Measure projection quality: how much information does rank-k SVD preserve?
+
+    For each N, tests multiple target_rank values. Reports:
+    - Energy ratio: fraction of total singular value energy in top-k
+    - Reconstruction error: projected vs full SVD
+    - Subspace agreement: cosine of principal angles between subspaces
+    - Timing: projected vs full SVD
+    """
+    print(f"\n{'='*100}")
+    print(f"  PROJECTION QUALITY ANALYSIS — B={B}, M={M}")
+    print(f"  Question: can rank-k SVD approximate rank-N SVD?")
+    print(f"{'='*100}")
+
+    configs = [
+        # (N, [target_ranks to test])
+        (32,  [8, 12, 16, 24]),
+        (48,  [8, 12, 16, 24, 32]),
+        (64,  [8, 12, 16, 24, 32, 48]),
+        (96,  [8, 16, 24, 32, 48, 64]),
+        (128, [8, 16, 24, 32, 48, 64, 96]),
+    ]
+
+    all_results = []
+
+    for N, ranks in configs:
+        if N > M:
+            continue
+
+        print(f"\n  N={N}:")
+        print(f"  {'k':>5}  {'Energy%':>8}  {'Recon_proj':>11}  {'Recon_trunc':>12}"
+              f"  {'S_rel_err':>10}  {'Subspace':>9}  {'Proj ms':>10}  {'Full ms':>10}  {'Speedup':>8}")
+        print(f"  {'─'*96}")
+
+        A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+
+        # Time full SVD once
+        full_ms, _, _ = _cuda_timer(lambda: gram_eigh_svd(A), warmup=10, iters=40)
+
+        for k in ranks:
+            if k >= N:
+                continue
+
+            q = projected_svd_quality(A, target_rank=k)
+            proj_ms, _, _ = _cuda_timer(
+                lambda: projected_svd(A, target_rank=k), warmup=10, iters=40)
+
+            speedup = full_ms / (proj_ms + 1e-9)
+
+            print(f"  {k:>5}  {q['energy_ratio']*100:>7.2f}%  {q['recon_proj']:>11.2e}"
+                  f"  {q['recon_trunc']:>12.2e}  {q['s_rel_err']:>10.4f}"
+                  f"  {q['subspace_cos']:>9.4f}  {proj_ms:>8.3f}ms  {full_ms:>8.3f}ms"
+                  f"  {speedup:>7.1f}x")
+
+            all_results.append({
+                'N': N, 'k': k, 'B': B, 'M': M,
+                'energy_ratio': round(q['energy_ratio'], 6),
+                'recon_proj': round(q['recon_proj'], 8),
+                'recon_trunc': round(q['recon_trunc'], 8),
+                's_rel_err': round(q['s_rel_err'], 6),
+                'subspace_cos': round(q['subspace_cos'], 6),
+                'proj_ms': round(proj_ms, 4),
+                'full_ms': round(full_ms, 4),
+            })
+
+        del A; torch.cuda.empty_cache()
+
+    # Summary table
+    print(f"\n  {'─'*70}")
+    print(f"  SUMMARY: Recommended target_rank per N")
+    print(f"  (≥99% energy, ≥0.99 subspace cos, best speedup)")
+    print(f"  {'─'*70}")
+    for N, ranks in configs:
+        good = [r for r in all_results if r['N'] == N
+                and r['energy_ratio'] >= 0.99 and r['subspace_cos'] >= 0.99]
+        if good:
+            best = min(good, key=lambda r: r['k'])
+            print(f"    N={N:>3}: k={best['k']:>3} → {best['energy_ratio']*100:.1f}% energy,"
+                  f" subspace={best['subspace_cos']:.4f},"
+                  f" {best['full_ms']/best['proj_ms']:.1f}x speedup")
+        else:
+            # Find best available
+            available = [r for r in all_results if r['N'] == N]
+            if available:
+                best = max(available, key=lambda r: r['energy_ratio'])
+                print(f"    N={N:>3}: best k={best['k']:>3} → {best['energy_ratio']*100:.1f}% energy,"
+                      f" subspace={best['subspace_cos']:.4f} (below 99% threshold)")
+
+    return all_results
+
+
+def profile_batch_sweep(N=3, M=1024):
+    """Sweep batch size for a fixed N. Shows scaling behavior."""
+    print(f"\n{'='*70}")
+    print(f"  BATCH SWEEP — N={N}, M={M}")
+    print(f"{'='*70}")
+    print(f"  {'B':>6}  {'Auto ms':>10}  {'Torch ms':>10}  {'Speedup':>8}  {'img/s':>12}")
+    print(f"  {'─'*52}")
+
+    batch_sizes = [32, 64, 128, 256, 512, 1024, 2048, 4096, 8192]
+    results = []
+
+    for B in batch_sizes:
+        try:
+            A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+        except RuntimeError:
+            print(f"  {B:>6}  OOM")
+            break
+
+        auto_mean, _, _ = _cuda_timer(lambda: batched_svd(A, method='auto'))
+        torch_mean, _, _ = _cuda_timer(
+            lambda: torch.linalg.svd(A, full_matrices=False))
+
+        speedup = torch_mean / (auto_mean + 1e-9)
+        ips = B / (auto_mean / 1000)
+
+        print(f"  {B:>6}  {auto_mean:>8.3f}ms  {torch_mean:>8.3f}ms  {speedup:>7.2f}x  {ips:>11,.0f}")
+        results.append({'B': B, 'N': N, 'M': M,
+                        'auto_ms': round(auto_mean, 4), 'torch_ms': round(torch_mean, 4),
+                        'speedup': round(speedup, 3)})
+        del A; torch.cuda.empty_cache()
+
+    return results
+
+
+def profile_spatial_sweep(N=3, B=512):
+    """Sweep spatial dimension M for a fixed N. Shows tiling efficiency."""
+    print(f"\n{'='*70}")
+    print(f"  SPATIAL SWEEP — N={N}, B={B}")
+    print(f"{'='*70}")
+    print(f"  {'M':>6} {'~HxW':>8}  {'Auto ms':>10}  {'Torch ms':>10}  {'Speedup':>8}")
+    print(f"  {'─'*48}")
+
+    m_values = [16, 64, 256, 512, 1024, 2048, 4096, 8192, 16384]
+    results = []
+
+    for M in m_values:
+        A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+        hw = int(M**0.5)
+        tag = f"{hw}×{hw}" if hw * hw == M else f"{M}"
+
+        auto_mean, _, _ = _cuda_timer(lambda: batched_svd(A, method='auto'))
+        torch_mean, _, _ = _cuda_timer(
+            lambda: torch.linalg.svd(A, full_matrices=False))
+
+        speedup = torch_mean / (auto_mean + 1e-9)
+        print(f"  {M:>6} {tag:>8}  {auto_mean:>8.3f}ms  {torch_mean:>8.3f}ms  {speedup:>7.2f}x")
+        results.append({'M': M, 'N': N, 'B': B,
+                        'auto_ms': round(auto_mean, 4), 'torch_ms': round(torch_mean, 4),
+                        'speedup': round(speedup, 3)})
+        del A; torch.cuda.empty_cache()
+
+    return results
+
+
+def profile_crossover_detail(M=1024, B=512):
+    """Fine-grained N sweep around expected crossover points."""
+    print(f"\n{'='*70}")
+    print(f"  CROSSOVER DETAIL — B={B}, M={M}")
+    print(f"{'='*70}")
+    print(f"  {'N':>4}  {'Gram ms':>10}  {'Torch ms':>10}  {'Winner':>8}  {'Margin':>8}")
+    print(f"  {'─'*46}")
+
+    for N in range(2, 65):
+        if N > M:
+            break
+        A = torch.randn(B, M, N, device="cuda", dtype=torch.float32)
+        gram_mean, _, _ = _cuda_timer(lambda: gram_eigh_svd(A), warmup=10, iters=40)
+        torch_mean, _, _ = _cuda_timer(
+            lambda: torch.linalg.svd(A, full_matrices=False), warmup=10, iters=40)
+
+        winner = "gram" if gram_mean < torch_mean else "torch"
+        margin = abs(gram_mean - torch_mean) / min(gram_mean, torch_mean) * 100
+
+        print(f"  {N:>4}  {gram_mean:>8.3f}ms  {torch_mean:>8.3f}ms  {winner:>8}  {margin:>6.1f}%")
+        del A; torch.cuda.empty_cache()
+
+
+# ╔═══════════════════════════════════════════════════════════════════════════╗
+# ║  MAIN                                                                   ║
+# ╚═══════════════════════════════════════════════════════════════════════════╝
+
+def main():
+    """Full profiling suite."""
+    assert torch.cuda.is_available(), "CUDA required"
+    device_name = torch.cuda.get_device_name(0)
+    print(f"{'='*80}")
+    print(f"  Generalized Batched Thin SVD — Profiling Suite")
+    print(f"  Device: {device_name}")
+    print(f"{'='*80}")
+
+    # Correctness first
+    run_validation(B=64, M=1024)
+
+    # Procrustes alignment quality — THE REAL QUESTION
+    # Does rank-k Procrustes produce the same rotation as rank-N?
+    procrustes_results = profile_procrustes_quality()
+
+    # Projection quality analysis — energy/reconstruction perspective
+    proj_results = profile_projection_quality(B=256, M=1024)
+
+    # N dimension sweep — timing comparison
+    n_results = profile_n_sweep(B=512, M=1024)
+
+    # Skip batch/spatial/crossover sweeps by default — uncomment if needed
+    batch_results = {}
+    spatial_results = {}
+    # for N in [3, 8, 32, 64]:
+    #     batch_results[N] = profile_batch_sweep(N=N, M=1024)
+    # for N in [3, 16, 48]:
+    #     spatial_results[N] = profile_spatial_sweep(N=N, B=512)
+    # profile_crossover_detail(M=1024, B=512)
+
+    # Summary
+    print(f"\n{'='*80}")
+    print(f"  SUMMARY")
+    print(f"{'='*80}")
+    print(f"\n  Strategy by N:")
+    print(f"    N=2:      Fused Triton (closed-form Jacobi rotation)")
+    print(f"    N=3:      Fused Triton (cyclic Jacobi in registers)")
+    print(f"    N=4-32:   Gram + eigh (bmm + cuSOLVER eigh) — sub-ms")
+    print(f"    N=48+:    Projected SVD (N→k, cheap SVD, lift back) — check quality table")
+    print(f"")
+    print(f"  Standalone utilities:")
+    print(f"    newton_schulz_invsqrt(G)              — batched G^{{-1/2}} via pure bmm")
+    print(f"    projected_svd(A, target_rank=k)       — rank-k approximate SVD")
+    print(f"    projected_svd_quality(A, target_rank)  — measure approximation quality")
+    print(f"")
+    print(f"  Key question answered: energy_ratio and subspace_cos in quality table")
+
+    # Save results
+    report = {
+        'device': device_name,
+        'procrustes_quality': procrustes_results,
+        'projection_quality': proj_results,
+        'n_sweep': n_results,
+        'batch_sweeps': {str(k): v for k, v in batch_results.items()},
+        'spatial_sweeps': {str(k): v for k, v in spatial_results.items()},
+    }
+    with open('svd_general_profile.json', 'w') as f:
+        json.dump(report, f, indent=2)
+    print(f"\n  Results saved to svd_general_profile.json")
+    print(f"{'='*80}")
+
+
+if __name__ == "__main__":
+    main()