Compute pairwise distance in a batch without replicating tensor in Tensorflow?

Computing pairwise distances in a batch within TensorFlow is a common task for measuring similarity or dissimilarity between samples in machine learning. To achieve this, we can use tensor operations to avoid extra tensor copying, thereby saving memory and improving computational efficiency.

Specifically, we can leverage TensorFlow's broadcasting mechanism and basic linear algebra operations. The following steps and example code illustrate how to compute pairwise Euclidean distances in a batch without copying tensors:

Steps

1. Determine the input tensor structure - Assume an input tensor X with shape [batch_size, num_features].

2. Compute squares - Use tf.square to square each element in X.

3. Compute sums - Use tf.reduce_sum to sum all features for each sample, resulting in a tensor of shape [batch_size, 1] representing the squared norm for each sample.

4. Compute squared differences using broadcasting - Exploit broadcasting to expand the shapes of X and the squared norm tensor to compute the squared differences between any two samples.

5. Compute Euclidean distances - Take the square root of the squared differences to obtain the final pairwise distances.

Example Code

python
import tensorflow as tf

def pairwise_distances(X):
    # Step 2: Calculate squared elements
    squared_X = tf.square(X)
    
    # Step 3: Reduce sum to calculate the squared norm
    squared_norm = tf.reduce_sum(squared_X, axis=1, keepdims=True)
    
    # Step 4: Compute squared differences by exploiting broadcasting
    squared_diff = squared_norm + tf.transpose(squared_norm) - 2 * tf.matmul(X, X, transpose_b=True)
    
    # Step 5: Ensure non-negative and compute the final Euclidean distance
    squared_diff = tf.maximum(squared_diff, 0.0)
    distances = tf.sqrt(squared_diff)
    
    return distances

# Example usage
X = tf.constant([[1.0, 2.0], [4.0, 6.0], [7.0, 8.0]])
print(pairwise_distances(X))

This code first computes the squared norms for each sample, then utilizes broadcasting to compute the squared differences between different samples, and finally calculates the pairwise Euclidean distances. This method avoids directly copying the entire tensor, thereby saving significant memory and improving computational efficiency when handling large datasets.