Functional Renormalization Group Approach for Signal Detection

Dear referees, dear editors,
We want to apologize for this delay in our response, which was largely be-
yond our control, and linked to the health problems of one of us. We hope you
will find our answers worthy of your expectations and patience. Furthermore,
we also want to thank the referee for his work, he took the time to read our
manuscript. Even though we did not agree with all of his remarks (which we
justify in what follows), this report also allowed us to improve the first version
and the presentation of our work.

About the part A

A1) It is not clear from the text what is shown in Fig.4. upper panel As I
understand the MP distribution is obtained from multiplying 2 fully random
matrices. How does the figure illustrate deviation from universality when the
difference could also be understood as a MP distribution with a different cutoff.
Answer A1: It’s difficult to illustrate this point on a figure, and one might
get this impression at first glance, but one of the major points highlighted
in our work is precisely that the renormalization group makes the difference!
At the most elementary level, since canonical dimensions are universally fixed
asymptotically (Figure 14). In contrast, the presence of a signal affects the
asymptotic value of these dimensions.

A2) A picture is introduced by which dimensionality is illustrated to be a decid-
ing factor in the relevance of eigenvectors because it induces different spectra
in momentum space. I think this is a dangerous analogy because dimension-
ality is not all that comes into play. Which eigendirections survive in the IR
limit depends on the field theory as well, so I do not understand what such an
analogy gains us.

Answer A2: This argument assumes that we know which field theory we are
talking about, or the number of fields involved in the construction of a given
interaction, for example. However, once this choice is given, the distribution
of moments is the only parameter deciding the relevance of the interactions.
Note that when we talk about "power counting", we generally assume that the
interactions are fixed, however, this showed us the importance of clarifying our
explanation, which we did (by a footnote).

About the part B

Beforehand, we want to remind you that this is not an article, but a review,
based on a series of published articles. Also, we do not claim to be exhaustive,
and certain points noted by the referee are still open questions. We asked
ourselves a lot of questions, we have so far dealt with a few, and this review
was an opportunity for a temporary assessment (requested by some of our
colleagues). In our writing, we have endeavored to follow the (non-definitive)
point of view adopted in the publications on which we base ourselves. Finally,
we also want to note that another article followed this review [2310.07499],
which has just been accepted into a journal, and which addresses yet other
aspects of the problem.

B1) My biggest problem with the paper is the 3rd section. I find it confusing and
vague, and I recommend rewriting it more simply and transparently, sacrificing
some of the material for clarity. There seem to be a lot of ideas there, however
as far as I see the only important point is left unanswered and this is: how does
looking at field theoretical models with RG help us detect signals in continuous
spectra? My point is this, when we are doing a renormalization group on a field
theoretical model such as the phi4 theory, we are at the end interested in low-
energy excitations of the problem. Why are these excitations relevant for signal
detection? Let’s look in comparison at the nicely introduced Wigner semicircle
spectrum and some peaks embedded in it. Nothing guarantees that what happens
in the low energy limit of the theory is relevant to detecting whatever peaks we
want to detect. So what I recommend to the authors is offering as simple as
possible answer to this question as the main point of Sec 3.

Answer B1 We do not understand the point of view of the referee. The signal,
following our investigation, is indeed found in the region of large eigenvalues
(this is also confirmed by the numerical and notably dimensional analyses of
sections 4 and 5), and is also found at the basis of ordinary signal detection
protocols like PCA. We could also say that this is a central hypothesis here,
since we are looking at small deformations around universality (and notable
deviations from the canonical dimension always occur on the side of large
eigenvalues).

B2) If the description of such signals as said in my previous point is not what
authors have in mind, they should be clear about what they are hoping to achieve
with their RG approach. What kind of signal is a signal that they are interested
in and that is analogous to the IR degrees of freedom that have survived after
the integration over small wave vector modes

Answer B2: Let us remember that this is a review, and the style relates
to it. This position has already been defended in our article, and the sec-
tion concretely shows some results which clearly illustrate our expectations.
Concretely, we want to know how the presence of a signal affects the univer-
sal properties of the flow, which is the content (as illustrated for example in
Figures 16-20-21).

B3) As far as I see later the authors introduce a phi4 like field theory with in
general a nonlocal kernel of interaction as we see in Eq. 3.4. which they then
discuss near the Gaussian limit and as the field theory teaches us to find that in
some relevant cases, the Gaussian theory is unstable and that higher couplings
above quadratic need to be taken into the account. What I find interesting is
that their theory is massive. Why is this the case or in other words why is this
case relevant for their consideration?

Answer B3: Considering a phi4 theory is essentially suggested by power
counting. Indeed, the theory is massive, and this is essentially due to the
interpretation we have of the value of mass for IR theory: it is the inverse of
the largest eigenvalue!

B4) Their interaction kernel was introduced as nonlocal. If this is the case,
why then do they only consider the standard derivative expansion of the gradi-
ent term in p2n? Why is their field theory not e.g. with long-range interactions?
In this case, they would have e.g. a fractional gradient term.

Answer B4: This is one of the project we are currently studying and be
released soon. This kind of theory would make it possible to explore the
interior of the "bulk", and not just the region of large eigenvalues. To build
our interactions, we added another hypothesis by working with the relative
variable p, and not p2. On this subject, the second part of section 3 is
an addition to the work mentioned in the reference articles, and we
placed as an appendix in this new version.

B5) Also why do they only consider the simplest situation of the scalar phi4
theory, let me elaborate. In 3.3 they discuss the symmetries of the model.
Why would they not have some more complicated order parameter field given
the symmetries?

Answer B5: We appreciate the referee’s comment of the referee, and this
question is also one of our concerns. We went as simple as possible for these
first investigations, which seems to be sufficient in the deep IR, but we are
currently studying the possibility of other symmetries.

About the part C
General remark: We come from the general remark made upstream by the
referee: The form of the calculations differs little from the standard case.
However, for the sake of pedagogy and to avoid any confusion about the small
specificity of the field theory that we are considering, we preferred to give
sufficient details to the reader.

C1) The difference between the standard calculation and their calculation should
be stressed better because it took me some time to understand that they are in-
deed not doing a standard phi4calculation.

Answer C1: We have added a note (in blue) at the beginning of the
section on this subject. However, we assume that the reader is not necessarily
aware of the formalism of the NPRG.

C2) If they claim that given their distribution of momenta the picture of the
flow is pretty much as in the standard case (which I think they do since evidence
to this effect is given later in sec 5), I do not understand what they mean by
their statement (*) above introduced as motivation. Do they mean that if they
tweak the couplings right that the dimensionless couplings are going to grow?
If this is so this statement has nothing to do with nonperturbative physics it
just has to do with the flow into the ordered phase. Even if you flow into the
disordered phase you are going to obtain some finite dimensionfull values for
the couplings when the flow stops. Authors please clarify this!

Answer C2: We initially had difficulty understanding the referee’s com-
ment before realizing that the error came from us! we wrote "nonperturba-
tive formalism" instead of "vertex expansion formalism". We rely on
the small number of relevant parameters in the IR to justify it, which would
not be possible in the UV, where the canonical dimensions explode! We have
modified the sentence.

C3) The authors seem to claim that there is no fixed point of their flow, how-
ever, Fig 21 testifies that there might be one. This is very peculiar. The
terminal part of the flow should be dominated by the shape momentum dis-
tribution near q=0. Why do the authors not discuss the asymptotic equations
separately? Considering these equations might give an analytic answer whether
there is a nontrivial fixed point or not in their case.

Answer C3: This is an infrared ambiguity! There are no fixed points (because
the canonical dimensions depend on the scale), but there can exist "fixed trajec-
tories", along which the flow cancels out. What we see in the figure is precisely
the terminal form of one of these lines. We have also discussed these “asymp-
totic fixed points” in a recent series [2403.07577, 2403.12217, 2404.11915].
We added a short remark, in blue, on the bottom of the Figure.

C4) My question is what kind of initial conditions do they have in mind when
they tell their statement?

Answer C4: Our assertion is only based on the power counting, therefore
the critical exponents in the vicinity of the Gaussian point, and does not claim
to describe the flow only in this vicinity. We agree with the remarks of the
referee, but it was not in question at this level. In general, all our dimensional
arguments are only valid in the vicinity of the Gaussian point.

C5) What is the relevance of the initial conditions of the flow to their
program of signal detection? In Sec 5 I seem to see that they are interested in
the critical situation. Why is that?

Answer C5: This question goes beyond the study carried out in this manuscript.
In our investigation, we have mainly focused on the Gaussian region for the
definition of our couplings, and we have been able to see a relation between
the intensity of the signal and the appearance of a symmetry breaking. In our
recent investigation, we explore the dependence of the detection threshold (the
moment when the purple region touches the blue region in Figure 25) on the
value of the initial scale (which plays the role of the UV cut-off in ordinary
field theory). We can then define an optimal threshold for detection, which
allows us to set a cut-off value between "signal" and "noise".

Finally, we checked the English throughout the manuscript and improved
the sentences.

On the Behalf of all the authors
Dr Samary