Systems and methods for machine learning

A computing system comprising at least one processor for comparing a sampling distribution and a target distribution, the computing system operable to:

• receive information defining the target distribution and a sampling process associated with the sampling distribution by the at least one processor;
• receive a plurality of samples by the at least processor, wherein the plurality of samples are generated using the sampling process associated with the sampling distribution;
• generate an empirical representation of the sampling distribution based on the plurality of samples by the at least one processor;
• generate a post-processed distribution from the plurality of samples by the at least one processor wherein to generate the post-processed distribution the at least one processor:
• determines a first and a second subset of variables;
• expresses a post-processed distribution in a mixture model comprising the sampling distribution and an analytic form for conditional resampling on the first and the second subset of variables;
• evaluates a Kullback-Leibler (KL) divergence of the target distribution on the post-processed distribution;
• and
• compares the sampling distribution to the target distribution based at least in part on the KL divergence.

A hybrid computing system comprising at least one digital processor and an analog processor for comparing a sampling distribution and a target distribution, the hybrid computing system operable to:

• receive information defining the target distribution and a sampling process associated with the sampling distribution by the at least one digital processor;
• receive a plurality of samples by the at least one digital processor, wherein the plurality of samples are generated by the analog processor using the sampling process associated with the sampling distribution;
• generate an empirical representation of the sampling distribution based on the plurality of samples by the at least one digital processor;
• generate a post-processed distribution from the plurality of samples by the at least one digital processor wherein to generate the post-processed distribution the at least one digital processor:
• determines a first and a second subset of variables;
• expresses a post-processed distribution in a mixture model comprising the sampling distribution and an analytic form for conditional resampling on the first and the second subset of variables;
• evaluates a Kullback-Leibler (KL) divergence of the target distribution on the post-processed distribution by the at least one digital processor;
• and
• compares the sampling distribution to the target distribution based at least in part on the KL divergence by the at least one digital processor.

A method of operation of a computational system comprising at least one processor, the method comprising:

• receiving, by at least one digital processor, information defining a target distribution and a sampling process associated with a sampling distribution;
• receiving, by at least one digital processor, a plurality of samples, the plurality of samples generated by at least one analog processor;
• generating, by the at least one digital processor, an empirical representation of the sampling distribution based on the received plurality of samples;
• generating, by the at least one digital processor, a post-processed distribution from the received plurality of samples, by:
• determining a first and a second subset of variables;
• expressing a post-processed distribution in a mixture model comprising the sampling distribution and an analytic form for conditional resampling on the first and the second subset of variables;
• evaluating a Kullback-Leibler (KL) divergence of the target distribution on the post-processed distribution by the at least one digital processor;
• and
• comparing the sampling distribution to the target distribution based at least in part on the KL divergence.

A method for generative machine learning by a computational system comprising a quantum processor and a digital processor, the method comprising:

• for a first iteration of a first iterative loop:
• determining, by the quantum processor, a first set of spin configurations for which the scalar product of a first weighting vector and a feature vector attains a largest value;
• and
• updating the first weighting vector, by the digital processor, to form an updated weighting vector based at least in part on a difference between the feature vector evaluated at the first set of spin configurations and a vector of feature moments;
• and
• for a subsequent iteration of the first iterative loop:
• determining, by the quantum processor, a subsequent set of spin configurations for which the scalar product of a subsequent weighting vector and the feature vector attains a largest value;
• and
• updating the subsequent weighting vector, by the digital processor, based at least in part on a difference between the feature vector evaluated at the subsequent set of spin configurations and the vector of feature moments.

A method for discriminative machine learning by a computational system comprising a quantum processor and a digital processor, the method comprising:

• for a first iteration of a first iterative loop:
• performing a first iteration of a second iterative loop, the second iterative loop nested in the first iterative loop, the second iterative loop over a plurality of sets of observed visible variables, the first iteration of the second iterative loop comprising:
• determining, by the quantum processor, a first set of labels for which the scalar product of a first weighting vector and the feature vector for a first set of the plurality of sets of observed visible variables attains a largest value;
• updating the first weighting vector, by the digital processor, to form an updated weighting vector based at least in part on an average discrepancy between the feature vector and a vector of feature moments, the average discrepancy determined over the plurality of sets of observed visible variables;
• for a subsequent iteration of the first iterative loop:
• performing a second iteration of the second iterative loop comprising:
• determining, by the quantum processor, a second set of labels for which the scalar product of a subsequent weighting vector and the feature vector for a second set of the plurality of sets of observed visible variables attains a largest value;
• updating the subsequent weighting vector, by the digital processor, to form an updated weighting vector based at least in part on an average discrepancy between the feature vector and a vector of feature moments, the average discrepancy determined over the plurality of sets of observed visible variables;
• and
• outputting at least one of the weighting vectors.