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.
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.
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
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:
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