Update No-Intercept Regression Slopes Across Batches

Data arrives in M batches. Each x_batches[b] and y_batches[b] is a matrix with the same number N of columns and the same number of rows within batch b. Column j of x is paired with column j of y.

For each cumulative prefix of batches, fit every paired no-intercept regression

$$ y_j=\beta_j x_j $$

using all rows seen so far. Implement:

cumulative_slopes(
 x_batches: List[Matrix[float]],
 y_batches: List[Matrix[float]]
) -> List[List[float]]

Output M rows of N slopes, where output row b uses batches 0 through b and

$$ \beta_j=\frac{\sum x_{ij}y_{ij}}{\sum x_{ij}^2}. $$

Constraints

• 1 <= M <= 1_000
• 1 <= N <= 500
• The total number of input rows across batches is at most 200_000.
• Every matrix is rectangular and paired batch shapes match.
• Every input value is finite and has absolute value at most 10^6.
• Each x value is either zero or has absolute value at least 10^-6.
• For every cumulative prefix and column, the denominator is positive.
• Floating-point answers within 1e-9 absolute or relative error are accepted.

Clarifications

• Do not include an intercept or center the data.
• A cumulative result uses all prior batches, not only the current batch.
• Batch row counts may differ.
• These bounds keep each product at most 10^12, each cumulative numerator and denominator at most 2 * 10^17 in magnitude, and every positive denominator at least 10^-12, preventing overflow or a non-finite slope in double precision.

Examples

Input:
x_batches = [[[1], [2]], [[3]]]
y_batches = [[[2], [4]], [[3]]]

Output:
[[2.0], [19.0 / 14.0]]

Evaluation Focus

• Correct column pairing and cumulative updates.
• Reuse of sufficient statistics rather than refitting over all history.
• (O(RN)) total time for R input rows and (O(N)) working state beyond the output.
• Stable floating-point accumulation and shape validation.

Extension

Which additional running statistics are needed to support an intercept?