AI Magazine - Spring 2018

terns, W↓m is the result of shifting the rows of the W matrix down m rows, and filling the displaced rows with zeros. This represents the basis patterns with a constant offset in the log q domain, and is equivalent to the original multiplicative shift in the q domain. The columns of Hm act as the activation of basis patterns for the basis patterns shifted down m units. Note that when M = 1, this formulation is equivalent to NMF, aside from the log transformation.

We can think of AgileFD as a family of algorithms that can be adapted to use different loss functions, regularization, and certain imposed constraints. Equation 2 is adapted from convolutive NMF, which was first proposed to analyze audio signals (Smaragdis 2004). The phase-mapping problem differs from previous applications of cNMF for blind source separation as the log q domain is substituted for the time domain, and each source (phase) is expected to appear at most once per sample with a relatively small offset. As in cNMF, AgileFD uses a gradient descent approach to fit W and H. When a generalized Kullback-Leibler (KL) divergence is used in the objective function, gradient updates can be written multiplicatively, and are applied iteratively until convergence. See Xue et al. (2017) for further details.

Lightweight Update Rules AgileFD’s linear gradient update rules solutions typically converge within minutes. This is orders of magnitude faster than CombiFD, which uses a similar problem formulation but with combinatorial constraints explicitly enforced globally, using a mixed-integer programming (MIP) representation. This increased efficiency of AgileFD enables high-throughput analysis and also makes it possible for a human to interact with the system in almost real time.

Further Extensions of AgileFD for Materials Discovery The ultimate aim of the phase-mapping problem is to find a physically meaningful decomposition of the signal. In the next few sections, we provide a number of novel modifications to the basic AgileFD algorithm, in order to impose prior knowledge or additional constraints derived from user interpretation of a proposed solution.

AgileFD with Frozen Values In the Phase-Mapper platform, the user is provided with the opportunity to freeze individual values in the W and H matrices. For example, a user might specify a known pattern or part of a previous solution as a basis pattern a priori, freezing the corresponding row or part thereof of W. Or the user might specify that a certain set of samples contain only a single phase and set the corresponding H values to zero. The result is interactive, iterative matrix factorization.

Custom Initialization By initializing basis patterns or coefficients to values close to the expected solution, rather than random values, the user can direct the search to the correct solution space. We allow the user to specify basis patterns that can be taken from previous solutions, from data samples, or be provided manually, to use as an initial value. Similarly, initial values for the activation matrix can be specified.

Sparsity Regularization Sparse solutions are more consistent with the underlying physics and are also usually more easily interpreted. The Phase-Mapper system provides the option to introduce a soft penalty term for sparsity in H, which can vary by index according to a human expert’s preferences.

The Gibbs Phase Rule
In general, correct solutions to the phase-mapping
problem should follow the Gibbs phase rule, which
specifies that the number of observed phases at a giv-
en chemical composition is no more than the num-
ber of chemical elements Nel:
( 4)

Here, Ii,j is an indicator of whether phase i is present at sample location j. Materials scientists might also know a priori, or infer from previous proposed solutions, that certain regions contain fewer phases than the usual limit.

Such combinatorial constraints cannot be encoded directly in the update rules of AgileFD, which has been used in previous methods such as CombiFD. However, these encodings result in a slow update process, as we have to solve a MIP problem in each iteration. As a novel routine, we apply the Gibbs phase rule by first solving the relaxed problem, then choose the best values to set to zero in H, and then refine the solution by applying the update rules until convergence. Because the update rules are multiplicative, the zeroed values will remain zero.

The value to set to zero in H is independent for each sample point j. This can be solved greedily if a faster solution is desired, or using a MIP formulation, which results in a small MIP program, or successive rounds of constraints and refinement, if a more precise solution is desired with a not much longer wait time. In addition, in the presence of alloying, the number of possible phases is further reduced by one. This alloying rule can also be captured with a small MIP program. These extensions are particularly useful when the unconstrained algorithm recovers a solution that is nearly correct except for relatively small violations of phase limits. See details in Bai et al. (2017).