Tsallis statistics for artificial intelligence, built on JAX.

Quickstart • Building blocks • Example • Installation

What is qjax?

Tsallis (non-extensive) statistics generalizes Boltzmann–Gibbs–Shannon statistics through a single entropic index $q$. As $q \to 1$ every construction collapses back to its classical counterpart — Shannon entropy, the Gaussian, softmax, the Kullback–Leibler divergence — while $q \neq 1$ opens up heavy tails, sparse attention, and tunable exploration.

qjax exposes these $q$-deformed primitives as pure, differentiable, jit/vmap-friendly JAX functions. Because $q$ is just another argument, you can hold it fixed or learn it end-to-end by gradient descent.

import jax, jax.numpy as jnp
import qjax

# q-deformed functions (recover log / exp as q -> 1)
qjax.q_log(2.0, q=1.5)
qjax.q_exp(1.0, q=1.5)

# Tsallis information measures
p = jnp.array([0.5, 0.3, 0.2])
qjax.tsallis_entropy(p, q=2.0)         # -> Shannon entropy as q -> 1
qjax.tsallis_divergence(p, p, q=2.0)   # -> KL divergence as q -> 1

# q-Gaussian distribution (heavy-tailed for 1 < q < 3)
x = jnp.linspace(-4, 4, 100)
qjax.q_gaussian_pdf(x, q=1.5, beta=1.0)
samples = qjax.sample(jax.random.PRNGKey(0), q=1.5, beta=1.0, shape=(1000,))

# Sparse softmax: q=1 -> softmax, q=2 -> sparsemax (exact zeros)
qjax.tsallis_entmax(jnp.array([2.0, 1.0, -1.0]), q=2.0)

Contents

• Building blocks
• A learnable q
• Label-noise robustness
• Installation
• Contributing
• License

Building blocks

qjax is organized as a small set of composable, fully differentiable primitives. Each is a pure function of $(x, q)$. Every primitive is a single closed form in the entropic index $q$, and each recovers its Boltzmann–Gibbs–Shannon counterpart in the $q \to 1$ limit:

qjax	Definition	Limit $q \to 1$
q_log	$\ln_q x = \dfrac{x^{1-q} - 1}{1 - q}$	$\ln x$
q_exp	$\exp_q x = \big[1 + (1-q),x\big]_+^{\frac{1}{1-q}}$	$e^{x}$
tsallis_entropy	$S_q(p) = \dfrac{1 - \sum_i p_i^{,q}}{q - 1}$	$-\sum_i p_i \ln p_i$
tsallis_cross_entropy	$H_q(y, p) = -\sum_i y_i \ln_q p_i$	$-\sum_i y_i \ln p_i$
tsallis_divergence	$D_q(p ,\Vert, r) = \dfrac{\sum_i p_i^{,q}, r_i^{,1-q} - 1}{q - 1}$	$\mathrm{KL}(p ,\Vert, r)$
q_gaussian_pdf	$\mathcal{G}_q(x) = \dfrac{\sqrt{\beta}}{C_q},\exp_q(-\beta x^2)$	$\sqrt{\tfrac{\beta}{\pi}},e^{-\beta x^2}$
tsallis_entmax	$entmax_q(z) = \arg\max_{p \in \Delta},\langle p, z\rangle + S_q(p)$	$softmax(z)$

where $[,\cdot,]_+ = \max(\cdot, 0)$ is the Tsallis cut-off, $C_q$ the $q$-Gaussian normalization, and $\Delta$ the probability simplex (tsallis_entmax is exactly sparsemax at $q = 2$).

qjax is a research library. The numerics are tested across the $q \to 1$ limit, gradients, and jit/vmap, but the API may still evolve.

Deformed functions and $q$-algebra

q_log and q_exp are inverse deformations of log/exp; the accompanying $q$-algebra turns them into homomorphisms (q_log(a·b) = q_add(q_log a, q_log b)).

qjax.q_log(x, q=1.5)                                   # (x**(1-q) - 1) / (1-q)
qjax.q_add(qjax.q_log(2.0, 1.4), qjax.q_log(3.0, 1.4), 1.4)   # == q_log(6.0, 1.4)

Information measures

p = jnp.array([0.5, 0.3, 0.2])
r = jnp.array([0.25, 0.25, 0.5])

qjax.tsallis_entropy(p, q=2.0)           # -> Shannon entropy as q -> 1
qjax.tsallis_cross_entropy(p, r, q=2.0)  # q-deformed cross-entropy loss
qjax.tsallis_divergence(p, r, q=2.0)     # -> KL(p || r) as q -> 1

The $q$-Gaussian

A maximum-Tsallis-entropy distribution: heavy-tailed (Student-$t$) for $1 < q < 3$, compactly supported for $q < 1$, Gaussian at $q = 1$.

x = jnp.linspace(-4, 4, 100)
qjax.q_gaussian_pdf(x, q=1.5, beta=1.0)
qjax.q_gaussian_logpdf(x, q=1.5, beta=1.0)
qjax.sample(jax.random.PRNGKey(0), q=1.5, beta=1.0, shape=(1000,))

Sparse activations

tsallis_entmax interpolates between dense softmax ($q = 1$) and sparsemax ($q = 2$), producing exact zeros for $q > 1$ — a drop-in for sparse attention.

z = jnp.array([2.0, 1.0, 0.1, -1.0])
qjax.tsallis_entmax(z, q=1.0)   # softmax (dense)
qjax.tsallis_entmax(z, q=2.0)   # sparsemax (exact zeros)

A learnable $q$

Because $q$ is an ordinary differentiable argument, it is finite everywhere — including the $q = 1$ limit — so it can be optimized like any other parameter:

import jax

x = jnp.linspace(-3, 3, 200)
nll = lambda q: -jnp.mean(qjax.q_gaussian_logpdf(x, q, 1.0))
grad_q = jax.grad(nll)(1.5)     # well-defined gradient w.r.t. the entropic index

This is what makes $q$ more than a hyperparameter: the right amount of non-extensivity can be discovered from data.

Label-noise robustness

When training labels are noisy, ordinary softmax cross-entropy is unbounded — a confidently mislabeled example incurs an arbitrarily large loss, so an over-parameterized network ends up memorizing the noise. Replacing the logarithm with the deformed $q$-logarithm gives the Tsallis cross-entropy, which is bounded for $q < 1$: its gradient saturates on unfittable points, so the model ignores label noise instead of fitting it.

For a one-hot target with true class $c$ and softmax probabilities $p$,

$$\mathcal{L}_q(p, c) = -\ln_q p_c = \frac{1 - p_c^{,1-q}}{1 - q}, \qquad \ln_q x = \frac{x^{1-q} - 1}{1 - q}.$$

As $q \to 1$ this is exactly the standard cross-entropy $-\log p_c$; for $q < 1$ the per-example loss is bounded above by $1/(1-q)$, so mislabeled points cannot dominate the gradient.

The figure trains a small 3-class classifier on two shapes (blobs, spiral) from clean data up to 40% label noise, comparing the Boltzmann–Gibbs–Shannon baseline ($q = 1$) with Tsallis ($q = 0.3$). The comparison is fair — both share the same initialization, data, noisy labels and optimizer; only $q$ differs. Without noise the two match (≈98–99%); as noise grows the baseline carves spurious wrong-class islands while Tsallis keeps clean regions and higher accuracy.

See the classification example for the full setup.

Installation

qjax requires Python 3.10+ and depends only on jax and matplotlib. It is managed with uv.

Use case	Command
As a dependency	uv add qjax
Development	uv sync --extra dev (tests + linter)
Building the docs	uv sync --extra docs (Sphinx + Furo)

For GPU/TPU acceleration, install the matching JAX build by following the JAX installation guide.

Contributing

Contributions are welcome — new $q$-deformed primitives, examples, docs, and fixes. See CONTRIBUTING.md for the development setup, design principles (purity, the $q \to 1$ limit, finite gradients), and the checks CI runs.

License

Released under the MIT License.