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The checker\nmaterializes `Q = torch.linalg.householder_product(H, tau)`, uses\n`R_factor = triu(H)`, and validates the LAPACK-style QR factorization\nresidual `R_factor - Q.T @ A` and orthogonality of `Q`. Since `R_factor`\nis extracted with `triu`, triangularity is part of the factorization check:\nif `Q.T @ A` has meaningful lower-triangular leakage, then it cannot match\n`R_factor`. The checker reports that lower-triangular leakage and the\nreconstruction residual as diagnostics.\n\nThis shape set targets optimizer-style matrix statistics where gradients are\nviewed as `[for_each..., basis_dim, contracted_dim]`, statistics are formed as\n`G @ G.T`, and QR is run on square `basis_dim x basis_dim` matrices. Batched\n`512 x 512` is especially important, while `1024`, `2048`, and `4096` cover\nlarger square factors.\n\nTest and benchmark specs include a `cond` field. In this task `cond` is a\ndeterministic input-scaling knob, not an exact requested condition number:\ndense cases multiply columns by `logspace(0, -cond, n)`, so larger `cond`\ncreates a wider dynamic range across columns. Some stress cases use their own\nstructure, such as rank-deficient, near-rank-deficient, banded, row-scaled,\nnear-collinear, upper-triangular, or clustered-scale inputs.\n\nThe `mixed` case builds a heterogeneous batch: each matrix is independently\nassigned a conditioning profile (a well-conditioned dense majority interleaved\nwith the ill-conditioned stress structures above) at a random position in the\nbatch. This mirrors the real optimizer-statistics regime, where the per-layer\nor per-block factors batched into one call have widely varying conditioning,\nrather than all sharing one structure. The benchmark set (not just the test\nset) now includes both `mixed` batches and fully ill-conditioned homogeneous\nbatches, so conditioning robustness is ranked, not only gated: an\nimplementation cannot inspect a few matrices, decide the whole batch is\nwell-conditioned, and route it to a path that is only valid for well-conditioned\ninputs, and the runtime cost of the accurate path on hard inputs is part of the\nscore. Each matrix must be factored correctly on its own merits.\n\nCorrectness is a hard gate against the original FP32 input and the FP32\n`torch.geqrf` compact-factor contract. Low-bit FP16, FP8, or NVFP4 work is\nallowed only as an internal implementation strategy: returned factors must\nstill be FP32 and must satisfy the same QR invariants as an FP32\nfactorization. Residuals are measured in FP64 to reduce checker noise, but\nthe target tolerance is still FP32 accuracy. The numerical property tolerance\nis purely relative, with no QR `atol`.\nThe hard gates are the LAPACK-style factor residual, which uses\n`rtol = 20 * n * eps32`, and orthogonality, which uses\n`rtol = 100 * n * eps32`, each applied to the corresponding matrix L1 norm.\nTriangularity is reported as lower-triangular leakage in `Q.T @ A` and is\nalready implied by the factor residual against `triu(H)`.\n\nAmong passing submissions, ranking is by runtime using the geometric mean of\nbenchmark cases. We will also celebrate notable submissions beyond the main\nleaderboard: the fastest, the most elegant, and the strangest working 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Paradise","validation_checked_at":null,"validation_contract_version":null,"validation_fully_validated":null,"validation_geomean_speedup":null,"validation_shapes_passed":null,"validation_shapes_total":null,"validation_status":null},{"file_name":"submission.py","line_count":6,"prev_score":0.20779495972760506,"rank":182,"score":0.4197366304351898,"submission_count":1,"submission_id":828432,"submission_time":"2026-06-22T21:02:39.086383+00:00","user_name":"Aakanksh Zarapkar","validation_checked_at":null,"validation_contract_version":null,"validation_fully_validated":null,"validation_geomean_speedup":null,"validation_shapes_passed":null,"validation_shapes_total":null,"validation_status":null},{"file_name":"baseline.py","line_count":6,"prev_score":0.003439031273483939,"rank":183,"score":0.4231756617086737,"submission_count":1,"submission_id":828200,"submission_time":"2026-06-22T18:03:20.601153+00:00","user_name":"Mohit Acharya","validation_checked_at":null,"validation_contract_version":null,"validation_fully_validated":null,"validation_geomean_speedup":null,"validation_shapes_passed":null,"validation_shapes_total":null,"validation_status":null}]},"reference":"import torch\nfrom task import input_t, output_t\n\n\n_FACTOR_RTOL_FACTOR = 20.0\n_ORTH_RTOL_FACTOR = 100.0\n\n\ndef _apply_column_scaling(a: torch.Tensor, cond: int) -> torch.Tensor:\n    # `cond` is a deterministic dynamic-range knob, not an exact condition number.\n    if cond:\n        n = a.shape[-1]\n        scales = torch.logspace(0.0, -float(cond), n, device=a.device, dtype=torch.float32)\n        return a * scales\n    return a.contiguous()\n\n\ndef _band_mask(n: int, bandwidth: int, device: torch.device) -> torch.Tensor:\n    idx = torch.arange(n, device=device)\n    return (idx[:, None] - idx[None, :]).abs() <= bandwidth\n\n\n# Per-matrix conditioning profiles drawn for the \"mixed\" case. \"dense\" is the\n# well-conditioned majority; the rest are the ill-conditioned stress structures.\n_MIXED_PROFILES = (\"dense\", \"rankdef\", \"nearrank\", \"clustered\", \"band\", \"rowscale\", \"nearcollinear\")\n# Relative sampling weights (normalized by torch.multinomial); dense ~= 50%.\n_MIXED_WEIGHTS = (6.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)\n\n\ndef _apply_case(a: torch.Tensor, case: str, cond: int, gen: torch.Generator) -> torch.Tensor:\n    # Apply one conditioning profile to an already-drawn base batch `a` of shape\n    # (m, n, n), drawing any case-specific extra randomness from `gen`. Factored\n    # out of generate_input so the homogeneous cases and the per-matrix \"mixed\"\n    # case share a single implementation. The draw order (base first in the\n    # caller, then the case extras here) matches the original code, so every\n    # homogeneous case produces bit-for-bit identical data to before.\n    m, n = a.shape[0], a.shape[-1]\n    device = a.device\n    if case == \"dense\":\n        a = _apply_column_scaling(a, cond)\n    elif case == \"upper\":\n        diag_boost = torch.linspace(1.0, 0.25, n, device=device, dtype=torch.float32)\n        a = torch.triu(a)\n        a.diagonal(dim1=-2, dim2=-1).add_(diag_boost)\n        a = _apply_column_scaling(a, cond)\n    elif case == \"diagonal\":\n        diag = torch.randn((m, n), device=device, dtype=torch.float32, generator=gen)\n        diag = diag.sign().clamp(min=0.0).mul(2.0).sub(1.0) * torch.logspace(\n            0.0, -float(max(cond, 2)), n, device=device, dtype=torch.float32\n        )\n        a = torch.diag_embed(diag)\n    elif case == \"rankdef\":\n        rank = max(1, (3 * n) // 4)\n        a[:, :, rank:] = 0.0\n        a = _apply_column_scaling(a, cond)\n    elif case == \"nearrank\":\n        rank = max(1, (3 * n) // 4)\n        tail = n - rank\n        if tail > 0:\n            noise = torch.randn(\n                (m, n, tail), device=device, dtype=torch.float32, generator=gen\n            )\n            a[:, :, rank:] = a[:, :, :tail] + 1.0e-5 * noise\n        a = _apply_column_scaling(a, cond)\n    elif case == \"clustered\":\n        scales = torch.ones((n,), device=device, dtype=torch.float32)\n        scales[n // 2 :] = 4.0 * torch.finfo(torch.float32).eps\n        if n >= 8:\n            lo = max(0, n // 2 - 2)\n            hi = min(n, n // 2 + 2)\n            scales[lo:hi] = torch.sqrt(torch.tensor(torch.finfo(torch.float32).eps, device=device))\n        a = a * scales\n    elif case == \"band\":\n        bandwidth = max(2, min(32, n // 32))\n        a = a * _band_mask(n, bandwidth, device)\n        diag_boost = torch.linspace(1.0, 0.5, n, device=device, dtype=torch.float32)\n        a.diagonal(dim1=-2, dim2=-1).add_(diag_boost)\n        a = _apply_column_scaling(a, cond)\n    elif case == \"nearcollinear\":\n        base = torch.randn((m, n, 1), device=device, dtype=torch.float32, generator=gen)\n        noise = torch.randn((m, n, n), device=device, dtype=torch.float32, generator=gen)\n        a = base.expand(m, n, n) + 1.0e-4 * noise\n        a = _apply_column_scaling(a, cond)\n    elif case == \"rowscale\":\n        row_cond = max(cond, 4)\n        scales = torch.logspace(0.0, -float(row_cond), n, device=device, dtype=torch.float32)\n        a = scales.reshape(1, n, 1) * a\n    else:\n        raise ValueError(f\"unknown QR test case: {case}\")\n    return a\n\n\ndef _generate_mixed(a: torch.Tensor, cond: int, gen: torch.Generator) -> torch.Tensor:\n    # Heterogeneous batch: assign each matrix an independent conditioning profile\n    # at a RANDOM position in the batch (seeded, so still deterministic), so\n    # well- and ill-conditioned matrices are interleaved rather than uniform\n    # across the batch. This matches the real optimizer-statistics regime (the\n    # per-layer / per-block factors have wildly different conditioning) and it\n    # removes the loophole where a kernel samples a few matrices, concludes the\n    # whole batch is well-conditioned, and routes it all to a fast path that is\n    # only numerically valid for well-conditioned inputs. With a mix present,\n    # passing the correctness gate requires handling each matrix on its merits.\n    m = a.shape[0]\n    device = a.device\n    weights = torch.tensor(_MIXED_WEIGHTS, dtype=torch.float32, device=device)\n    labels = torch.multinomial(weights, m, replacement=True, generator=gen)\n    # Guarantee both a well-conditioned and an ill-conditioned matrix are present.\n    # (Only relevant for tiny batches; large batches get both with high prob.)\n    if m >= 2:\n        is_dense = labels == 0\n        if not bool(is_dense.any()):\n            labels[int(torch.randint(0, m, (1,), device=device, generator=gen))] = 0\n        elif bool(is_dense.all()):\n            pos = int(torch.randint(0, m, (1,), device=device, generator=gen))\n            labels[pos] = int(torch.randint(1, len(_MIXED_PROFILES), (1,), device=device, generator=gen))\n    # Process profiles in fixed order over the present labels so the RNG draws\n    # inside _apply_case are deterministic for a given seed.\n    for k, prof in enumerate(_MIXED_PROFILES):\n        mask = labels == k\n        if bool(mask.any()):\n            a[mask] = _apply_case(a[mask], prof, cond, gen)\n    return a\n\n\ndef generate_input(batch: int, n: int, cond: int, seed: int, case: str = \"dense\") -> input_t:\n    assert batch > 0, \"batch must be positive\"\n    assert n > 0, \"n must be positive\"\n    assert cond >= 0, \"cond must be non-negative\"\n\n    device = \"cuda\" if torch.cuda.is_available() else \"cpu\"\n    gen = torch.Generator(device=device)\n    gen.manual_seed(seed)\n\n    case = case.lower()\n    a = torch.randn((batch, n, n), device=device, dtype=torch.float32, generator=gen)\n\n    if case == \"mixed\":\n        a = _generate_mixed(a, cond, gen)\n    else:\n        a = _apply_case(a, case, cond, gen)\n\n    return a.contiguous()\n\n\ndef ref_kernel(data: input_t) -> output_t:\n    # Starter/reference path: correctness first; submissions compete on speed.\n    return torch.geqrf(data)\n\n\ndef _property_rtol(n: int, factor: float) -> float:\n    eps = torch.finfo(torch.float32).eps\n    return factor * max(n, 1) * eps\n\n\ndef _scaled_residual(\n    residual: torch.Tensor,\n    scale: torch.Tensor,\n    n: int,\n) -> torch.Tensor:\n    eps = torch.finfo(torch.float32).eps\n    return residual / (eps * max(n, 1) * scale.clamp_min(1e-30))\n\n\ndef _matrix_l1_norm(value: torch.Tensor) -> torch.Tensor:\n    return torch.linalg.matrix_norm(value.double(), ord=1, dim=(-2, -1))\n\n\ndef _check_tensor(name: str, value: torch.Tensor, shape: tuple[int, ...], device: torch.device) -> str | None:\n    if not isinstance(value, torch.Tensor):\n        return f\"{name} must be a torch.Tensor\"\n    if value.shape != shape:\n        return f\"{name} shape must be {shape}, got {tuple(value.shape)}\"\n    if value.dtype != torch.float32:\n        return f\"{name} dtype must be torch.float32, got {value.dtype}\"\n    if value.device != device:\n        return f\"{name} must be on {device}, got {value.device}\"\n    if not torch.isfinite(value).all().item():\n        return f\"{name} contains NaN or Inf\"\n    return None\n\n\ndef check_implementation(data: input_t, output: output_t) -> tuple[bool, str]:\n    a = data\n    batch, n, _ = a.shape\n    factor_rtol = _property_rtol(n, _FACTOR_RTOL_FACTOR)\n    orth_rtol = _property_rtol(n, _ORTH_RTOL_FACTOR)\n\n    if not isinstance(output, tuple) or len(output) != 2:\n        return False, \"output must be a tuple `(H, tau)`\"\n\n    h, tau = output\n    error = _check_tensor(\"H\", h, (batch, n, n), a.device)\n    if error is not None:\n        return False, error\n    error = _check_tensor(\"tau\", tau, (batch, n), a.device)\n    if error is not None:\n        return False, error\n\n    q = torch.linalg.householder_product(h, tau)\n    r = torch.triu(h)\n    if not torch.isfinite(q).all().item():\n        return False, \"Q materialized from `(H, tau)` contains NaN or Inf\"\n    if not torch.isfinite(r).all().item():\n        return False, \"R extracted from `triu(H)` contains NaN or Inf\"\n\n    a_check = a.double()\n    q_check = q.double()\n    r_check = r.double()\n    projected = q_check.transpose(-1, -2) @ a_check\n    if not torch.isfinite(projected).all().item():\n        return False, \"Q.T @ A contains NaN or Inf\"\n\n    factor_residual = _matrix_l1_norm(r_check - projected)\n    factor_scale = _matrix_l1_norm(a_check)\n    factor_allowed = factor_rtol * factor_scale\n    factor_scaled = _scaled_residual(factor_residual, factor_scale, n)\n    if not torch.isfinite(factor_scaled).all().item():\n        return False, \"R - Q.T @ A residual produced NaN or Inf\"\n    factor_failed = factor_residual > factor_allowed\n    if bool(factor_failed.any().item()):\n        worst = int(factor_scaled.argmax().item())\n        return False, (\n            \"R - Q.T @ A is too large: \"\n            f\"matrix={worst}, residual={factor_residual[worst].item():.3g}, \"\n            f\"allowed={factor_allowed[worst].item():.3g}, \"\n            f\"scaled={factor_scaled[worst].item():.3g}\"\n        )\n\n    eye = torch.eye(n, device=a.device, dtype=torch.float64).expand(batch, n, n)\n    qtq = q_check.transpose(-1, -2) @ q_check\n    if not torch.isfinite(qtq).all().item():\n        return False, \"Q.T @ Q contains NaN or Inf\"\n    orth_residual = _matrix_l1_norm(qtq - eye).amax()\n    orth_scale = _matrix_l1_norm(eye).amax()\n    orth_allowed = orth_rtol * orth_scale\n    orth_scaled = _scaled_residual(orth_residual, orth_scale, n)\n    if not torch.isfinite(orth_scaled).all().item():\n        return False, \"Q.T @ Q residual produced NaN or Inf\"\n    if orth_residual.item() > orth_allowed.item():\n        return False, (\n            \"Q is not orthogonal enough: \"\n            f\"residual={orth_residual.item():.3g}, allowed={orth_allowed.item():.3g}, \"\n            f\"scaled={orth_scaled.item():.3g}\"\n        )\n\n    lower = torch.tril(projected, diagonal=-1)\n    tri_residual = _matrix_l1_norm(lower).amax()\n    tri_scale = _matrix_l1_norm(a_check).amax()\n    tri_scaled = _scaled_residual(tri_residual, tri_scale, n)\n\n    recon = q_check @ r_check\n    if not torch.isfinite(recon).all().item():\n        return False, \"Q @ R contains NaN or Inf\"\n    recon_residual = _matrix_l1_norm(recon - a_check).amax()\n    recon_scale = _matrix_l1_norm(a_check).amax()\n    recon_scaled = _scaled_residual(recon_residual, recon_scale, n)\n\n    return True, (\n        f\"factor_rtol={factor_rtol:.3g}; \"\n        f\"orth_rtol={orth_rtol:.3g}; \"\n        f\"scaled_factor_residual={factor_scaled.amax().item():.3g}; \"\n        f\"scaled_reconstruction_residual={recon_scaled.item():.3g}; \"\n        f\"scaled_triangular_residual={tri_scaled.item():.3g}; \"\n        f\"scaled_orthogonality_residual={orth_scaled.item():.3g}; \"\n        f\"batch={batch}; n={n}\"\n    )\n","starter":"import torch\nfrom task import input_t, output_t\n\n\ndef custom_kernel(data: input_t) -> output_t:\n    return torch.geqrf(data)\n","time_left":"ended"},"message":"Success"}