Function log_softmax_simd 

pub fn log_softmax_simd<F>(x: &ArrayView1<'_, F>) -> Array1<F>
where
    F: Float + SimdUnifiedOps,

Apply Log-Softmax function using SIMD operations

The log-softmax function is defined as: log_softmax(x_i) = x_i - log(Σ_j exp(x_j))

This is more numerically stable than computing log(softmax(x)) directly, and is critical for neural network training:

• Cross-entropy loss computation
• Classification networks (output layer)
• Transformer language models
• Large language models (LLMs)

Mathematical Properties

• log_softmax(x) = x - logsumexp(x)
• Σ exp(log_softmax(x)) = 1 (outputs are log-probabilities)
• log_softmax(x + c) = log_softmax(x) (invariant to constant shift)
• Maximum output element approaches 0 for peaked distributions

Arguments

• x - Input array of logits (unnormalized log-probabilities)

Returns

• Array with log-softmax applied elementwise (log-probabilities that sum to 0)

Example

use scirs2_core::ndarray_ext::elementwise::log_softmax_simd;
use ndarray::{array, ArrayView1};

let logits = array![1.0_f64, 2.0, 3.0];
let log_probs = log_softmax_simd(&logits.view());

// Verify: exp(log_probs) sums to 1
let probs: f64 = log_probs.mapv(|lp| lp.exp()).sum();
assert!((probs - 1.0).abs() < 1e-10);

// log_softmax values are always <= 0
for &lp in log_probs.iter() {
    assert!(lp <= 0.0);
}

Applications

• Cross-Entropy Loss: -Σ target * log_softmax(logits)
• Classification: Converting logits to log-probabilities
• Transformers: Final output layer for token prediction
• Language Models: Computing perplexity and token probabilities
• Numerical Stability: Avoiding underflow in softmax computation