The ability to learn and compose functions is foundational to efficient
learning and reasoning in humans, enabling flexible generalizations such as
creating new dishes from known cooking processes. Beyond sequential chaining of
functions, existing linguistics literature indicates that humans can grasp more
complex compositions with interacting functions, where output production
depends on context changes induced by different function orderings. Extending
the investigation into the visual domain, we developed a function learning
paradigm to explore the capacity of humans and neural network models in
learning and reasoning with compositional functions under varied interaction
conditions. Following brief training on individual functions, human
participants were assessed on composing two learned functions, in ways covering
four main interaction types, including instances in which the application of
the first function creates or removes the context for applying the second
function. Our findings indicate that humans can make zero-shot generalizations
on novel visual function compositions across interaction conditions,
demonstrating sensitivity to contextual changes. A comparison with a neural
network model on the same task reveals that, through the meta-learning for
compositionality (MLC) approach, a standard sequence-to-sequence Transformer
can mimic human generalization patterns in composing functions.

在研究中，我们通过探索人类和神经网络模型在学习和推理组合函数方面的能力，发现人类能够在不同的交互条件下对视觉函数组合进行零样本泛化，比较同一任务中的神经网络模型发现，通过元学习组合性（MLC）方法，标准序列到序列的转换器能够模仿人类在函数组合中的泛化模式。

人类和机器的功能组合学习

Compositional learning of functions in humans and machines

We prove an impossibility result, which in the context of function learning
says the following: under certain conditions, it is impossible to
simultaneously learn symmetries and functions equivariant under them using an
ansatz consisting of equivariant functions. To formalize this statement, we
carefully study notions of approximation for groups and semigroups. We analyze
certain families of neural networks for whether they satisfy the conditions of
the impossibility result: what we call ``linearly equivariant'' networks, and
group-convolutional networks. A lot can be said precisely about linearly
equivariant networks, making them theoretically useful. On the practical side,
our analysis of group-convolutional neural networks allows us generalize the
well-known ``convolution is all you need'' theorem to non-homogeneous spaces.
We additionally find an important difference between group convolution and
semigroup convolution.

使用等变函数作为认知模型的假设条件下，学习具有对称性和等变性的函数是不可能的；我们探究了群和半群的逼近概念，分析了线性等变网络和群卷积网络是否满足该结果，并阐述了它们的理论和实际意义。

等变模型的普适性与对称性的可学习性之间的权衡

A tradeoff between universality of equivariant models and learnability  of symmetries

In classical machine learning, regression is treated as a black box process
of identifying a suitable function from a hypothesis set without attempting to
gain insight into the mechanism connecting inputs and outputs. In the natural
sciences, however, finding an interpretable function for a phenomenon is the
prime goal as it allows to understand and generalize results. This paper
proposes a novel type of function learning network, called equation learner
(EQL), that can learn analytical expressions and is able to extrapolate to
unseen domains. It is implemented as an end-to-end differentiable feed-forward
network and allows for efficient gradient based training. Due to sparsity
regularization concise interpretable expressions can be obtained. Often the
true underlying source expression is identified.

该研究提出了一种名为等式学习器（EQL）的新型函数学习网络，可以学习解析表达式，并能够外推到未知的领域。通过稀疏正则化可以得到简洁明了的可解释表达式，可用于高效的基于梯度的训练。