Logsoftmaxix

Logsoftmaxix is a term used to describe the log-softmax operation applied along a designated axis of a tensor, where ix indicates the axis index. It is the log-probability version of the softmax function, computed separately for each slice of the tensor along that axis.

Mathematically, for a tensor X and a chosen axis dim = ix, the output Y has the same shape as X, with elements given by Y = log_softmax(X, dim=ix). In practical terms, for each fixed set of indices other than dim, the values along axis dim are normalized to form a probability distribution, and the logarithm of these probabilities is returned. Concretely, for a one-dimensional input, logsoftmaxix(x) reduces to the standard log-softmax: log(exp(x_i) / sum_j exp(x_j)) for each i.

Numerical stability is a key consideration. The common and stable implementation uses the log-sum-exp trick: log_softmax(x) = x - log(sum_exp(x)) along the chosen axis, where sum_exp(x) is computed with care to avoid overflow. In practice, many deep learning libraries expose this as a parameterized function, allowing the user to specify the axis along which normalization occurs.

Applications and usage include computing log-probabilities for classification tasks, sequence models, and attention mechanisms where normalization needs to occur along a specific dimension. It is also a building block in loss functions such as cross-entropy, where working in log-probabilities helps improve numerical stability.

Relation to related concepts: logsoftmaxix is closely aligned with softmax and log_softmax, differing mainly by the axis of operation. Properly specifying the axis ensures correct normalization across multi-dimensional inputs. See also: softmax, log_softmax, cross-entropy.