We investigate how machine learning models acquire the ability to compose music and how musical information is internally represented within such models. We develop a composition algorithm based on a restricted Boltzmann machine (RBM), a simple generative model capable of producing musical pieces of arbitrary length. We convert musical scores into piano-roll image representations and train the RBM in an unsupervised manner. We confirm that the trained RBM can generate new musical pieces; however, by analyzing the model's responses and internal structure, we find that the learned information is not stored in a form directly interpretable by humans. This study contributes to a better understanding of how machine learning models capable of music composition may internally represent musical structure and highlights issues related to the interpretability of generative models in creative tasks.

We introduce theoretically grounded Continuous Semi-Quantum Boltzmann Machines (CSQBMs) that supports continuous-action reinforcement learning. By combining exponential-family priors over visible units with quantum Boltzmann distributions over hidden units, CSQBMs yield a hybrid quantum-classical model that reduces qubit requirements while retaining strong expressiveness. Crucially, gradients with respect to continuous variables can be computed analytically, enabling direct integration into Actor-Critic algorithms. Building on this, we propose a continuous Q-learning framework that replaces global maximization by efficient sampling from the CSQBM distribution, thereby overcoming instability issues in continuous control.

We extend the multinomial logit model to represent some of the empirical phenomena that are frequently observed in the choices made by humans. These phenomena include the similarity effect, the attraction effect, and the compromise effect. We formally quantify the strength of these phenomena that can be represented by our choice model, which illuminates the flexibility of our choice model. We then show that our choice model can be represented as a restricted Boltzmann machine and that its parameters can be learned effectively from data. Our numerical experiments with real data of human choices suggest that we can train our choice model in such a way that it represents the typical phenomena of choice.

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models.

Quantum computers offer the potential for efficiently sampling from complex probability distributions, attracting increasing interest in generative modeling within quantum machine learning. This surge in interest has driven the development of numerous generative quantum models, yet their trainability and scalability remain significant challenges. A notable example is a quantum restricted Boltzmann machine (QRBM), which is based on the Gibbs state of a parameterized non-commuting Hamiltonian. While QRBMs are expressive, their non-commuting Hamiltonians make gradient evaluation computationally demanding, even on fault-tolerant quantum computers. In this work, we propose a semi-quantum restricted Boltzmann machine (sqRBM), a model designed for classical data that mitigates the challenges associated with previous QRBM proposals. The sqRBM Hamiltonian is commuting in the visible subspace while remaining non-commuting in the hidden subspace. This structure allows us to derive closed-form expressions for both output probabilities and gradients. Leveraging these analytical results, we demonstrate that sqRBMs share a close relationship with classical restricted Boltzmann machines (RBM). Our theoretical analysis predicts that, to learn a given probability distribution, an RBM requires three times as many hidden units as an sqRBM, while both models have the same total number of parameters. We validate these findings through numerical simulations involving up to 100 units. Our results suggest that sqRBMs could enable practical quantum machine learning applications in the near future by significantly reducing quantum resource requirements.

We extend the multinomial logit model to represent some of the empirical phenomena that are frequently observed in the choices made by humans. These phenomena include the similarity effect, the attraction effect, and the compromise effect. We formally quantify the strength of these phenomena that can be represented by our choice model, which illuminates the flexibility of our choice model. We then show that our choice model can be represented as a restricted Boltzmann machine and that its parameters can be learned effectively from data. Our numerical experiments with real data of human choices suggest that we can train our choice model in such a way that it represents the typical phenomena of choice.

Detailed comments: 40 - You might be interested in [MacKay, David. Artificial Intelligence] is one earlier example of its use that I'm familiar with. A paper by Salakhutdinov suggests that [H. Stat., 22:400-407, 1951.] is the earliest reference, though I have not read this older paper. Doesn't m have a single fixed point?

Generative models aim to create samples statistically similar to those belonging to a training dataset: their goal is to fit the probability distribution from which the datapoints supposedly come from. In a generic setting, this probability distribution takes the form of a Boltzmann factor. The corresponding Energy Based Models (EBMs) fit the parameters of the Hamiltonian of the Boltzmann distribution and can be viewed as maximum entropy models, where the statistical properties of the dataset are imposed as constraints to low degree correlation functions [1, 2], (see [3, 4] for recent reviews). The resulting learning rule can be viewed as a gradient ascent on the Log-Likelihood (LL). However, running the training dynamics is a notoriously challenging task: at each epoch, the evaluation of the gradient of the LL requires the computation of the correlation functions of the degrees of freedom as predicted from the current estimation of the model's probability distribution. This is typically an intractable problem from an analytical point of view and is generally tackled numerically through parallel Monte Carlo Markov Chain (MCMC) simulations.

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.