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The authors have declared that no competing interests exist.

It was recently shown that quantum annealing can be used as an effective, fast subroutine in certain types of matrix factorization algorithms. The quantum annealing algorithm performed best for quick, approximate answers, but performance rapidly plateaued. In this paper, we utilize reverse annealing instead of forward annealing in the quantum annealing subroutine for nonnegative/binary matrix factorization problems. After an initial global search with forward annealing, reverse annealing performs a series of local searches that refine existing solutions. The combination of forward and reverse annealing significantly improves performance compared to forward annealing alone for all but the shortest run times.

Due to the slowing progress of classical computing [

The NBMF algorithm factors a matrix

One downside of the quantum annealer approach is that improvement in solution quality from iteration to iteration quickly plateaus. This is because the forward annealing approach that was used previously could only perform global searches when solving the binary least squares problem. This ignores the results of the solutions from previous iterations, which is likely a good starting point for the next iteration. Instead, the annealing process almost always produces a factor matrix that is very different from the factor matrix at the previous iteration. In practice, this means that the algorithm hops around solution space at random, quickly finding good solutions but never refining them beyond a certain level of accuracy.

Fortunately, the latest iteration of the D-Wave hardware, the 2000Q, allows us to explore solutions around some initial classical state. This process is known as reverse annealing. In this paper, we utilize reverse annealing to improve performance of the NBMF algorithm. Specifically, we use reverse annealing to explore local minima near an initial state defined by the results of the previous iteration of the algorithm. This significantly reduces the iteration-over-iteration change in the algorithm, allowing promising solutions to be refined rather than discarded.

The NBMF algorithm takes a real-valued

We will now give an outline of the NBMF algorithm and the implementation on the D-Wave; for full details of the algorithm see [^{(0)}, each iteration follows an alternating least squares approach:

Returning to our matrix factorization problem, the columns _{j} in

For consistency, we use the same data set (2,429 facial images) and rank

The original implementation of NBMF on the D-Wave used the standard forward anneal procedure, where the device starts in an equal superposition of all possible states. Our motivation for this study is to use a new feature of the D-Wave 2000Q, reverse annealing, in order to improve performance. Reverse annealing begins in a specified classical state, then explores solutions in the local vicinity of that initial state. This allows us to iteratively improve upon solutions from previous iterations of the algorithm, rather than conducting global searches at every step.

The NBMF algorithm begins with a random initialization of the

In the simplest terms, the reverse anneal process depends on two parameters:

reversal distance

reversal time _{r} (in

The (dimensionless) reversal distance _{r} controls how long the search is conducted. A longer search has a greater chance of returning a lower energy sample, but at the cost of slowing the algorithm down. For this analysis, we do not employ any other tuning of the D-Wave, e.g. spin reversal transforms.

For a given reversal distance _{r}, the anneal schedule we use is
_{r}, and then re-anneal the system. See _{r}, and the D-Wave default forward anneal schedule.

_{r}, while default forward anneal schedule (orange) increases from

In addition to specifying the reverse anneal schedule, we must also specify the initial state. As discussed in the introduction, the NBMF algorithm naturally provides an initial configuration based on the results of the previous iteration of the algorithm. Specifically, if we are solving for

We characterize the efficacy of a reverse anneal sample by seeing if it:

is the same as the initial state,

has a lower energy than the initial state (good),

has a higher energy than the initial state (bad).

The frequency with which samples fall in to each category gives us an idea of how effective the reverse anneal is at finding improved solutions.

We studied the effects of reversal time _{r} and reversal distance _{r} and _{r} does not significantly increase the likelihood of discovering better states. The peak probability of discovering a lower-energy sample was 13.5% ± 9.1% for _{r} = 10_{r} = 100_{r} = 10_{r} = 100_{r} = 10

Evaluated for 100 randomly chosen QUBOs appearing during an evaluation of the NBMF algorithm for with _{r} = 10_{r} = 100_{r} = 10_{r} = 100

We note that the choice of these parameters has some dependence on the matrix that is being factored. For example, the same calibration procedure evaluated on a matrix with random values (as opposed to the highly structured facial imagery data) revealed an optimal reversal distance of

There is additional computational overhead related to the reverse anneal process, such as configuring the hardware in to the chosen initial state before each anneal. Therefore, for the purposes of comparing forward and reverse anneal efficacy we will look at quality of solution vs. total QPU access time (as opposed to (annealing time × number of anneals), as was done in [

When comparing the reverse anneal results against the original forward anneal version of the algorithm, we allot each method equal QPU access time. Given the timing values discussed above, we find that the ratio

In this section we use reverse annealing in the NBMF algorithm to factor the dataset of 2,429 facial images studied in [

Mean performance reported from five evaluations of the forward and reverse annealing versions of the NBMF algorithm, with 1000 forward anneals and 240 reverse anneals per QUBO, corresponding to a total QPU access time of 6182 seconds over the full evaluation of each algorithm. Standard deviation (not shown) was less than 1% of the mean.

Recall that our hypothesis, outlined in the introduction, is that reverse annealing will outperform forward annealing due to more refinement of existing solutions as opposed to generation of entirely new solutions per QUBO. If we define
^{(i+1)} and ^{(i)} divided by the size of

Data taken during the evaluation of the algorithms as described in

In

Data taken from 725 distinct evaluations of the NBMF algorithm, varying the number of forward and reverse anneals per evaluation. Reverse annealing results in up to a 12% increase in performance.

The reason that forward annealing outperforms reverse annealing for small sample size is straightforward. For a single reverse anneal sample, the likelihood of finding a better state than the initial configuration is quite low (≤ 25%, sometimes much lower, see

We performed a benchmark using Gurobi that is similar to the benchmark performed previously for this problem [

The time required for Gurobi to find a solution that is as good as or better than the solution found by reverse annealing for each QUBO in a factorization problem is shown in comparison to the QPU access time and the annealing time. Cases where reverse annealing failed to find a better solution are excluded from the plot. In many cases, the time required by Gurobi exceeds both the annealing time and the QPU access time.

The results of this work suggest that reverse annealing improves the quality of the NBMF factorization by 12% for this application. This improvement is seen when the number of reverse anneals evaluated per QUBO is at least 7 (which is equivalent in QPU access time to 29 forward anneals). In [

In addition to characterizing the performance in terms of the quality of the factorization given a fixed time, it could be characterized in terms of how long it takes to obtain a factorization of a given quality. By this standard, reverse annealing would also perform well once the quality of the factorization is set sufficient low. Since NBFM with forward annealing tends to plateau at a worse factorization quality, the speed-up with reverse annealing would be very large once the factorization quality is set beyond this plateau.

Our results could be improved upon in several ways. First, it is possible that the optimal reverse anneal schedule could depend on how many iterations have already occured (i.e., as better solutions become harder to find). It is also our hope that future quantum annealing hardware will feature more rapid state initialization, as this accounts for over 98% of the additional time related to reverse annealing. This would improve the performance of NBMF with reverse annealing but leave the performance of NBMF with forward annealing unchanged. Lastly, the exact nature of the matrix being factorized appears to play a role in determining how effective the algorithm is, and this could be explored further.