Centered Gram / Covariance Fusion

This fusion targets covariance or Gram-matrix construction where a tall matrix is mean-centered, multiplied by its transpose, and scaled by n or n-1.

What Qualifies

• Pattern: mean(X, 1) → X - mean → (X - mean)' * (X - mean) → division by a scalar (n-1 or n).
• Transpose + mul pairing. One operand of the multiplication must be a transpose node whose input matches the centred matrix.
• Scalar divisor. The denominator must be a compile-time scalar constant so the planner can identify biased vs unbiased covariance.
• Exclusive ownership. All nodes involved (mean, subtraction, transpose, multiplication, division) must not already belong to another fusion group.

Why Centered Gram?

• Centered Gram is a common pattern in linear algebra and statistics. It shows up in covariance matrix construction, principal component analysis, and other linear algebra operations
• By fusing this pattern, we can keep the Gram matrix resident on the GPU, avoiding the overhead of uploading and downloading it to the CPU.

Benefits

• Zero host traffic. Neither the centred matrix nor the Gram matrix round-trips through the CPU.
• Fewer GPU launches. We can consolidate multiple covariance matrix constructions into a single kernel, reducing the number of GPU launches.

Known Limitations

• Only the canonical layout (mean → subtraction → transpose → matmul → scalar divide) is detected today.
• Weighted covariance or per-column scaling is not yet fused.
• Very large matrices may still trigger size guardrails in provider_impl.rs, forcing a fallback to the CPU implementation.

If your program follows the pattern above and still falls back, confirm that the divisor literal matches either size(X,1) or size(X,1)-1 exactly; otherwise the planner cannot classify the normalization.