
Submitted by
Assigned_Reviewer_4
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The paper proposes an approach for constructing a
linear wavelet transform on weighted graphs based on the lifting scheme,
which has a number of favourable properties: 1) linear in memory and time
with the size of the graph, 2) adaptive to a class of signals, 3) exact
analysis and synthesis, i.e. allows for perfect signal reconstruction, 4)
efficient training through resemblance with deep autoencoder networks.
The paper is presented well: it is clearly structured and well
written. After a nice overview and introduction, the authors give a
detailed derivation of their construction and show in a number of
experiments the validity and versatility of their approach.
The
proposed approach and wavelet construction builds on previous work but
makes a nontrivial contribution to the field of graphbased signal
processing by deriving a generalpurpose wavelet transform with a number
of favourable properties.
The authors make an interesting
connection between wavelet construction on graphs and autoencoder
networks. It is likely that this paper will trigger further development in
this line of research. It is also likely to serve as a flexible tool in
the analysis of signals on graphs.
Additional comments: *
great if the authors could be more precise what sufficient in section 4.5
means? In a general problem how would one determine how many eigenvectors
need to be taken into account? * What is the meaning of the colorbars
in Fig. 4 and 5. ? * In Sec. 4.7 change "It is also possible o" >
"It is also possible to"
Q2: Please summarize your
review in 12 sentences
The paper elaborates a nontrivial generalpurpose
wavelet transform for signals on weighted graphs, which exhibits a number
of favourable properties. It makes an interesting connection to
autoencoder networks and is likely to trigger further work along these
lines.
Submitted by
Assigned_Reviewer_5
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This work is aimed to provide interface between the
signal processing theory of wavelets and the deep neural network. What is
presented is only a small step toward this goal, but is interesting in
demonstrating the feasibility of the approach.
It is interesting
to see the various connections among wavelet construction and deep
autoencoder.
The detail is difficult to follow, and I hope the
presentation can be drastically improved to enhance the readability.
Q2: Please summarize your review in 12
sentences
interesting work to bridge signal processing theory of
wavelets and deep learning. But details are difficult to follow, and the
presentation should be drastically improved to enhance the
readability. Submitted by
Assigned_Reviewer_6
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
Summary: The authors present a method for constructing
wavelets on weighted graphs that can adapt to a class of signals. The
approach uses the lifting scheme and by connecting this scheme to a deep
autoencoder network, the authors are able to perform unsupervised
training similar to the pretraining of a stack of autoencoders. The
approach is linear in the size of the graph. The authors explore the
performance of the wavelet construction approach through application to
several data sets.
Quality: The authors provide an elegant
approach for taking into account the class of signals when forming
wavelets on a graph. Most constructions are based solely on the graph
structure, and those few methods that do allow adaptivity based on the
signals [19,21] have significant limitations.
The technical
development is clearly described, novel, and leads to an efficient
algorithm that produces wavelets with desirable properties.
The
partitioning approach is described very briefly and there is almost no
discussion of how sensitive the approach is to the partitioning method.
The spectral clustering algorithm of [20] has some limitations
(performance being poor for some kinds of graphs) and it would be nice to
see more information about how sensitive the overall wavelet construct is
to the partitioning scheme.
For their results on irregular graphs,
where I think the construction is of most interest, the authors do not
make a comparison to compression or reconstruction using any other type of
graph wavelet. Instead, the comparison is to (somewhat dated) learning
techniques that are suited to manifold analysis. This constitutes one
significant weakness of the paper.
Clarity: The paper is very well
written and the development is easy to follow. As discussed above, more
detail on some of the
Originality: The authors provide a new
method for constructing wavelets on graphs that has the important ability
to adapt to the class of functions that the wavelets will be used to
represent. The method is highly original.
Significance: The paper
represents a useful contribution in the field of wavelets and
multiresolution analysis on graphs, a field of growing interest due to the
numerous potential applications. I consider that the signal adaptivity,
while preserving linearity and efficiency of construction, represents a
significant advance.
Q2: Please summarize your
review in 12 sentences
The paper makes a significant original contribution,
providing an important advance in the field of wavelets on graphs. The
lack of a thorough comparison to other wavelet graphs for compression and
reconstruction is the major weakness of the paper.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank the reviewers for their thoughtful comments
and suggestions. We are happy that the reviews generally recognize the
interest and novelty of the graph wavelet construction introduced in the
paper and share our enthusiasm for further exploration of this area. Below
we respond to the main issues raised by the reviewers. Given the
opportunity, we will incorporate the necessary modifications into the
final version of the paper.
Partitioning: =========
Since our approach is based on improving upon the Haar wavelets, their
quality will have an effect on the final construction. As proved in
[Gavish et al], the quality of Haar wavelets depends on the quality of the
partitioning; c.f. Theorem 1 and formula (8). From practical standpoint,
it is hard to achieve high quality partitions on all types of graphs using
a single algorithm. For the graphs used in our paper, we have also run
experiments with METIS partitioning as in [Allard et al], and a
combination of METIS at coarse levels with spectral partitioning at finer
levels, with results similar to what is presented in the paper. Our choice
of spectral clustering is motivated by the facts that it has some
theoretical guarantees [Szlam] and blends nicely with the rest of
exposition. We will be glad to discuss the choice of partitioning in the
paper more thoroughly.
Comparison to other wavelets:
========= We sought to provide a comparison with previous wavelets
in a setting with established theory and state of the art algorithms. Our
example on the Yale Faces dataset provides such a direct comparison to the
state of the art wavelet filters used in JPEG 2000 standard with favorable
results. Note that the underlying pixel graph used in our construction is
not the standard grid, but each pixel (including the ones on the image
boundary) is connected to its 8 nearest neighbors.
If further
comparisons with existing graph wavelets are desirable, we are happy to
include them in the final version. However, in contrast to our approach,
the sparsity of representation has not been directly optimized for in
previous graph wavelet constructions; rather it was expected for smooth
functions as a consequence of the vanishing moment property. This
expectation comes from the setting of the real line, where the number of
vanishing moments is the main factor determining the asymptotic decay
speed of wavelet coefficients of smooth functions [Mallat, Section 6.1].
However, no generalization of this result is available in graph setting
except for Haar wavelets [Gavish et al]. More importantly, to the best of
our knowledge, it is not known how to define the notion of higher order
vanishing moments for wavelets on graphs in a way that will result in a
faster asymptotic decay than that of Haar wavelets. In other words, in
terms of sparsity of representation, the Haar wavelets are the state of
the art in this nascent field of signal processing on graphs. As stated in
[Shuman et al. 2013]: “A major open issue in the field of signal
processing on graphs is how to link structural properties of graph signals
and their underlying graphs to properties (such as sparsity and
localization) of the generalized operators and transform coefficients.
Such a theory could inform transform designs, and help identify which
transforms may be better suited to which applications.”
As for
our comparison to learning techniques, it is motivated by some early
attempts of utilizing graph wavelets for semisupervised learning. [Shuman
et al 2011] uses the spectral graph wavelets for semisupervised learning
by including a structured sparsity penalty, yet they find that their
prediction performance is sometimes slightly better and sometimes slightly
worse than methods based on global smoothness priors. They conclude:
“However, this is somewhat disappointing due to the significant additional
complexity of the proposed spectral graph wavelet method.” On the other
hand, our wavelets achieve improvements over the methods based on global
smoothness priors using only a simple sparsity penalty (L1 norm).
Number of eigenvectors to use. ========= The
number of training functions required to robustly train the neural
networks depends on the number of parameters; in our case this is related
to the number of the neighbors that a region can have at a given level. In
the cases discussed in the paper, graphs have a lowdimensional structure,
and the number of neighboring partitions is low  which allows the
training to succeed with a small number of training functions. For high
dimensional point clouds a larger number (growing with the intrinsic
dimension of the manifold) of training functions will be required.
Exposition ========= We will be glad to follow any
specific suggestions for improving the exposition of the paper. We will
certainly fix the typos and include an explanation of colorbars in
captions of Figs. 4 and 5.
References =========
[Allard et al.] W.K. Allard, G. Chen, M. Maggioni. Multiscale
Geometric Methods for Data Sets II: Geometric MultiResolution Analysis.
Appl. Comp. Harm. Anal., 32(3): 435462.
[Gavish et al.] Matan G.,
Boaz N., Ronald R. C.: Multiscale Wavelets on Trees, Graphs and High
Dimensional Data: Theory and Applications to Semi Supervised Learning.
ICML 2010: 367374.
[Mallat] S. Mallat. A Wavelet Tour of Signal
Processing, 2nd ed. Academic Press, 1999.
[Shuman et al. 2011]
Shuman, D. I.; Faraji, M. J.; Vandergheynst, P.: SemiSupervised Learning
with Spectral Graph Wavelets. SampTA 2011.
[Shuman et al. 2013] D.
I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst. The
emerging field of signal processing on graphs: Extending highdimensional
data analysis to networks and other irregular domains. IEEE Signal
Process. Mag., 30(3):83–98, 2013
[Szlam] A. Szlam. Asymptotic
regularity of subdivisions of Euclidean domains by iterated PCA and
iterated 2means. Appl. Comp. Harm. Anal., 27(3): 342350, 2009.
 