use log-sum-exp in estimate_log_prob_resp to prevent overflow (Ref to #443) (#444)

* Implement log-sum-exp trick in Gaussian Mixture Model to prevent overflow

* fix lint problem

Author: schliffen

algorithms/linfa-clustering/src/gaussian_mixture/algorithm.rs

@@ -341,10 +341,11 @@ impl<F: Float> GaussianMixtureModel<F> {
         observations: &ArrayBase<D, Ix2>,
     ) -> (Array1<F>, Array2<F>) {
         let weighted_log_prob = self.estimate_weighted_log_prob(observations);
-        let log_prob_norm = weighted_log_prob
-            .mapv(|x| x.exp())
-            .sum_axis(Axis(1))
-            .mapv(|x| x.ln());
+        // Log-sum-exp trick to avoid overflow: ln(Σ exp(xᵢ)) = max + ln(Σ exp(xᵢ - max))
+        let log_max = weighted_log_prob
+            .map_axis(Axis(1), |row| row.fold(F::neg_infinity(), |a, &b| a.max(b)));
+        let shifted = &weighted_log_prob - &log_max.clone().insert_axis(Axis(1));
+        let log_prob_norm = shifted.mapv(|x| x.exp()).sum_axis(Axis(1)).mapv(|x| x.ln()) + &log_max;
         let log_resp = weighted_log_prob - log_prob_norm.to_owned().insert_axis(Axis(1));
         (log_prob_norm, log_resp)
     }