Spin Boltzmann machine

Abstract

Boltzmann machine (BM) is a recurrent network, which has a wide range of applications in machine learning (ML) including dimensionality reduction, feature learning and classification. Standard BM is described by the Ising model and can be implemented as a spin ice based device. Such hardware implementation is faster and more energy efficient than a simulation on digital computers. At the moment, a hardware BM is a single purpose device designed on digital computers for a specific task. In the paper we propose a generalized BM capable of fitting parameters by demonstration of training examples, which is done completely inside the spintronic device. Our generalization is based on the Heisenberg model, which is more accurate than the Ising model for spin ice. We show that for some systems minimization of Kullback-Leibler divergence during training of BM is equivalent to minimization of free energy with respect to the biases of the units, hence training of the ML model can be done by energy dissipation. We include the biases as degrees of freedom of the device, whose dynamics is described by the same Landau-Lifschitz-Gilbert equation as for spins representing units of BM. The demonstration of samples from the training set is done by fixing inputs and outputs according to ground truth. The training samples are remembered by the machine becoming minima on the energy landscape implementing a kind of long-term potentiation. The performance of the proposed machine is compared with a single layer perceptron artificial neural network and with a Bernoulli restricted BM on a binary classification problem. KEYWORDS spectral gap, quantum graph, Schr ödinger operator, discrete spectrum.

Toward nanomagnetic implementation of energy-based machine learning

“…The optimization of the mean m is relatively simple and was already solved in [16]. Here we let m = 0 and focus on optimization of θ.…”

“…Our initiative is to implement BM loss minimization as a relaxation in a magnetic device. In the article [16], we proposed an extended BM machine whose weights are included to degrees of freedom (DoF). We showed that the energy minimization of the system leads to memorization of the trained samples.…”

On spin flip for electron scattering by several delta-potentials for 1D Hamiltonian with spin-orbit interaction

“…It is not the electron charge but the electron spin that carries information [1]. The new wave of interest was inspired by the development of quantum computing and quantum computations (see, e.g., [2,3]). Correspondingly, one needs electronic devices which can distinguish the spin orientation and can control it.…”