Page 220 - Contributed Paper Session (CPS) - Volume 4

CPS2201 Mikhail L.
treated active learning for the Cook - Torrance model Cook and Torrance
(1981). Analysis of BRDF data with statistical and machine learning methods
was discussed in Langovoy (2015b), Langovoy (2015a), Sole et al. (2018),
Doctor and Byers (2018).

2. Active learning and design of experiments
In general, BRDF is a 5-dimensional manifold, having 4 angular and 1
wavelength dimension. Note that even a set of 1-dimensional manifolds is
infinite-dimensional (and k-dimensional manifolds are not to be confused with
parametric −dimensional families of functions). At the same time, a typical
measuring device only takes between 50 and 1000 points for all the BRDF
layers together. In view of this, the available measurement points are indeed
very scarce for a complicated problem such as BRDF estimation. Therefore, an
efficient sampling strategy is required when performing the measurements.
Since sets of BRDF measurements are, in fact, observed random manifolds, we
are dealing here with manifold-valued data.
Statistical design of experiments (see Fisher et al. (1960), Cox and Reid
(2000)) is a well developed area of quantitative data analysis. However,
previous research in this field was often more concerned with (important)
topics such as manipulation checks, interactions between factors, delayed
effects, repeatability, among many others. This shifted the focus away from
considering design of statistical experiments on structured, constrained, or
infinite-dimensional data. In contrast, BRDF measurements are carried out in
strictly defined settings and by qualified experts. Therefore, there is less room
for human or random errors and influences. On the other hand, BRDF
measurements are collections of points representing manifolds, so defining
even the simplest statistical quantities in this case turns out to be a nontrivial
and conceptual task.
Overall, our methodology represents a far-reaching generalization of the
active machine learning framework, also generalizing the proactive learning
setup of Donmez and Carbonell (2008). Active learning, as a special case of
semi-supervised machine learning, oftentimes deals with finite sets of labels
and aims at solving classification or clustering problems with a finite number
of classes. While there have been a number of promising practical applications,
most of the existing theory deals with analysis of performance of specific
algorithms (query by committee, algorithm, or importance weighted
2
approach, among a few others) under rather restrictive conditions on the loss
functions, incoming distributions, and other components of the learning
model. For recent developments, we refer to Agarwal et al. (2013), Beygelzimer
et al. (2009), Dasgupta and Hsu (2008).

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