This paper improves existing methods that provide explanations using trainable masks. Inspired by methods for static data, the current approach uses fixed perturbations. However, this paper argues that this approach may not be suitable for time series data. In this paper, their method not only has a trainable mask but also trainable perturbations.

$$ \Phi(\mathbf{x}, \mathbf{m}) = \mathbf{m} \times \mathbf{x} + (1 - \mathbf{m}) \times g(\mathbf{x}), $$

where $g(\mathbf{x})$ is a function of the input. A possible definition is $g(\mathbf{x}) = \frac{1}{W}\sum_{t'=t-W}^{t}x_{t'}$.

$$ \Phi(\mathbf{x}, \mathbf{m}) = \mathbf{m} \times \mathbf{x} + (1 - \mathbf{m}) \times NN(\mathbf{x}) $$$$ \argmin_{\mathbf{m}, \Theta \in NN} \lambda_{1} \Vert \mathbf{m}\Vert_{1} + \lambda_{2} \Vert NN(\mathbf{x})\Vert_{1} + \mathcal{L}(f(\mathbf{x}), f(\Phi(\mathbf{x}, \mathbf{m}))) $$

they decompose the objective function as follows:

  • $\Vert \mathbf{m}\Vert_{1}$ induces $\Phi(\mathbf{x})$ to be closed to $NN(\mathbf{x})$
  • $\Vert\Phi(\mathbf{x})\Vert_{1}$ induces $\Phi(\mathbf{x})$ to be close $\mathbf{0}$ (uninformative)
  • $\mathcal{L}$ induces $f(\Phi(\mathbf{x}, \mathbf{m}))$ to be close to $f(\mathbf{x})$ (informative)

It is worth mentioning that, when applying the ‘deletion game’, this paper considers the $-\mathcal{L}(f(\mathbf{x}), f(\Phi(\mathbf{x}, \mathbf{m})))$ objective function difficult to optimize because making $f(\Phi(\mathbf{x}, \mathbf{m}))$ distant from $f(\mathbf{x})$ does not specify where the distant point should be. Therefore, they use $-\mathcal{L}(f(\mathbf{0}), f(\Phi(\mathbf{x}, \mathbf{m})))$ instead of the original objective function.

References

Enguehard, J. (2023, July). Learning perturbations to explain time series predictions. In International Conference on Machine Learning (pp. 9329-9342). PMLR.