Haufe Feature Importance (Activation Patterns) — haufe

Computes per-feature importance using the Haufe et al. (2014) transformation from decoding weights to encoding (activation) patterns: A = Sigma_x %*% W %*% solve(t(W) %*% Sigma_x %*% W).

Usage

haufe_importance(
  W,
  Sigma_x = NULL,
  summary_fun = function(A) sqrt(rowSums(A^2)),
  X = NULL,
  center = TRUE,
  block_size = NULL
)

Arguments

W: A P x D weight matrix (features x discriminant directions).
Sigma_x: A P x P covariance matrix of the training features. Ignored when X is supplied.
summary_fun: A function applied to the rows of A to produce a scalar importance per feature. Defaults to the L2 norm across discriminants.
X: Optional n x P matrix of training observations. When supplied, the matrix-free path is used and Sigma_x is not needed.
center: Logical; centre the columns of X before computing the patterns (default TRUE, matching cov()).
block_size: Optional integer; when X is supplied, compute the activation patterns in column blocks of at most this many features.

Value

A list with components:

A: The P x D activation pattern matrix.
importance: A numeric vector of length P with per-feature importance.

Details

Two equivalent computation paths are available. Supplying Sigma_x uses the explicit $P \times P$ covariance. Supplying the training observations X instead uses the matrix-free identity $$Z = X_c W$$ $$A = X_c^\top Z (Z^\top Z)^{+}$$ where $X_c$ is the column-centred data. The covariance normalisation cancels exactly, so the two paths agree to numerical precision, but the matrix-free path never forms a $P \times P$ matrix: it needs only $n \times D$, $P \times D$, and $D \times D$ quantities. Use it for whole-brain feature counts, where a dense covariance is infeasible. Because $Z$ is centred, $X_c^\top Z = X^\top Z$, so X itself is never copied; block_size additionally limits the number of feature columns multiplied at once.

References

Haufe, S., Meinecke, F., Goergen, K., Daehne, S., Haynes, J.D., Blankertz, B., & Biessmann, F. (2014). On the interpretation of weight vectors of linear models in multivariate neuroimaging. NeuroImage, 87, 96-110.

Examples

# \donttest{
  W <- matrix(rnorm(10*2), 10, 2)
  X <- matrix(rnorm(50*10), 50, 10)
  haufe_importance(W, cov(X))
#> $A
#>              [,1]        [,2]
#>  [1,] -0.20601881 -0.04836532
#>  [2,] -0.26969162 -0.07218315
#>  [3,]  0.02941763 -0.15261698
#>  [4,] -0.06307709  0.04272807
#>  [5,]  0.14622583 -0.27530518
#>  [6,] -0.06839436 -0.01409065
#>  [7,] -0.14964591  0.13610741
#>  [8,]  0.09851969 -0.19713427
#>  [9,] -0.25944046 -0.16014209
#> [10,]  0.17747513  0.08697459
#> 
#> $importance
#>  [1] 0.21161984 0.27918448 0.15542632 0.07618666 0.31172895 0.06983075
#>  [7] 0.20228476 0.22038159 0.30488496 0.19764109
#> 
  # identical, without forming cov(X):
  haufe_importance(W, X = X)
#> $A
#>              [,1]        [,2]
#>  [1,] -0.20601881 -0.04836532
#>  [2,] -0.26969162 -0.07218315
#>  [3,]  0.02941763 -0.15261698
#>  [4,] -0.06307709  0.04272807
#>  [5,]  0.14622583 -0.27530518
#>  [6,] -0.06839436 -0.01409065
#>  [7,] -0.14964591  0.13610741
#>  [8,]  0.09851969 -0.19713427
#>  [9,] -0.25944046 -0.16014209
#> [10,]  0.17747513  0.08697459
#> 
#> $importance
#>  [1] 0.21161984 0.27918448 0.15542632 0.07618666 0.31172895 0.06983075
#>  [7] 0.20228476 0.22038159 0.30488496 0.19764109
#> 
# }