This version March 16, 2018

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Maxiset point of view for signal detection in inverse problems
Florent Autin1 , Marianne Clausel2 , Jean-Marc Freyermuth3 , Clément Marteau4
March 16, 2018
1 3

Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France.
Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France,
4
Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622
Villeurbanne, France.

arXiv:1803.05875v1 [] 15 Mar 2018

2

1 Introduction
Over the last 20 years, the assessment of the performance of nonparametric function estimation
methods mainly relied on the asymptotic minimax and oracle approaches. More marginally
used, the maxiset paradigm has been proved to be very useful to accurately describe the behaviour of some estimation procedures. In some cases, it allows to distinguish methods having
comparable minimax performance. The question of adapting the maxiset concepts to the signal
detection framework was often raised. This is the aim of this paper to rigorously extend this
point of view to the signal detection framework and to discuss new related outcomes.
To this end, we will deal all along the paper with the Gaussian sequence space model
yk = bk θk + εξk ,

k ∈ J,

(1)

where (yk )k∈J denotes the observations, J ⊂ N∗ is a subset of N∗ , θ = (θk )k∈J a non negative
unknown sequence of interest, (bk )k∈J a given sequence of non negative real numbers, ε a noise
level in (0, 1) and (ξk )k∈J a sequence of i.i.d. standard Gaussian random variables. The model
(1) allows to describe several situations, as e.g. nonparametric regression or estimation of a
function blurred by white noise. For more details on these models and their connection with the
Gaussian sequence space model we refer the interested reader to Tsybakov (2009). For the sake
of convenience, we will consider hereafter that J = N∗ = N \ {0}. We also stress that the model
(1) allows to deal with so-called inverse problem models as described in Cavalier (2011). In
such a setting, one is interested in doing inference on a function f ∈ H in some Hilbert space H
from indirect and blurred observation of the form
Y = A f + εξ,

(2)

where A : H → K denotes a compact operator acting from H to another Hilbert space K, ε a
noise level and ξ a Gaussian white noise. In particular, the sequence (b2k )k∈N∗ can be identified
as the sequence of eigenvalues of the operator A⋆ A and the sequence (θk )k∈N∗ as the one of the
coefficients of f in the singular values decomposition (SVD) basis associated to the operator A.

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For inverse problem models, the minimax paradigm has been widely used in order to assess
the performance of estimation procedures. Roughly speaking, given a structural constraint on
the vector θ of interest, typically of the form θ ∈ Θ for some Θ ⊂ l2 (N∗ ), one measures the
performance of a given estimator θ̂ through its maximal risk
R(θ̂) = sup Eθ ℓ(θ, θ̂),
θ∈Θ

where ℓ(., .) denotes a given loss function. This paradigm has been widely used and discussed
over the years. In several situations, a precise bound can be obtained on R(θ̂) which allows to
characterize how the maximal risk decreases with respect to the noise level ε.
More precisely, one can often exhibit a non decreasing positive sequence (rε )ε>0 , the so–called
rate of convergence associated to the estimation procedure θ̂ with noise level ǫ, such that R(θ̂) ≤
Crε for some positive constant C. If this rate appears to be the smallest possible one, namely if
there exists a positive constant c such that
inf sup Eθ ℓ(θ, θ̃) ≥ crε ,
θ̃

θ∈Θ

the sequence (rε )ε>0 is called minimax rate of convergence over Θ. In the previous inequality,
the infimum is taken over all possible estimators θ̃ of θ. We refer, e.g., to Tsybakov (2009),
Johannes et al. (2011) for a non-exhaustive reference list.
Under the minimax estimation paradigm, the performance of two given procedures can be compared through their respective rates of convergence according to a chosen functional set Θ.
However, it does not always allow for comparison if both procedures are ’minimax-optimal’.
In addition the used criterium is quite pessimistic: the risk is measured at the slowest possible estimation precision over the set Θ. Hence, it does not provide a fair comparison. To
tackle these issues, an alternative point of view has been proposed in the seminal paper of
Kerkyacharian and Picard (2002). The main idea can be stated as follows; given an estimation
procedure θ̂ and a sequence of rates (rε )ε>0 , can we determine the set Θ MS (θ̂) of sequences θ than
are estimated by θ̂ at the rate (rε )ε>0 ? If yes, the set Θ MS (θ̂) is called the maxiset associated to
the procedure θ̂ for the rate (rε )ε>0 . Under this paradigm, the best performing procedure, i.e., the
’maxiset-optimal’ procedure, is the one whose associated maxiset strictly includes the maxisets
of the others. Note that a very usual criticism concerns the situation where estimation methods
have non nested maxisets. Autin et al. (2012) discuss this important aspect of the maxiset approach explaining that, first, it is is somehow normal to find that some estimation methods are
better in estimating some specific functions. In such a case, examining the ’form’ of the maxiset
will bring interesting information. Second, it may be possible to combine these procedures such
that the maxiset combined procedure contains the union of the maxisets. The maxiset point of
view has been generalized to various settings, see, e.g., Autin (2006), Rivoirard and Tribouley
(2008) or Hohage and Weidling (2017).

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In the framework of signal detection, the minimax point of view has been widely investigated and was very fruitfully applied. We refer to Ingster and Suslina (2003), Baraud (2002),
Ingster et al. (2012) or Laurent et al. (2012) among others. Nevertheless, as in the estimation
case, the minimax paradigm does not allow for a fully satisfying comparison between different
testing procedures. The extension of the maxiset theory to this setting is a doorway to novel informative and rigorous math-stat study of these procedures. A flavor of the maxiset approach in
signal detection framework has been discussed in Autin et al. (2014). Nevertheless, the proper
adaptation of this approach to the signal detection framework is a challenging problem that we
tackle in this paper. We further discuss some new issues related to this theory. In particular, in
this work, we aim at
• highlighting the link between the space Θ and the sequence (rε )ε>0 that both appear in
the alternative hypothesis of the testing problem for different procedures based on χ2 statistics,
• comparing inverse and direct approaches in the light of the maxiset point of view.
In Section 2, we recall the minimax paradigm and then present the maxiset point of view for
signal detection problems. Thereafter, in Section 3, we state maxiset results for both the inverse
approach and the direct one (see Theorems 3.1 and 3.2). A crucial and perhaps surprising
aspect in signal detection in inverse problem has been raised in Laurent et al. (2011) where
they compared direct and indirect testing procedures, ie., from the minimax point of view. In
Section 4 we succeed in comparing inverse and direct approaches in the light of the maxiset
approach in many cases (see Proposition 4.1) as for instance the moderately ill-posed inverse
problem (see Proposition 4.2).
Following a brief conclusion on the novelty of our results in Section 5, we postpone in Section 6
all the related proofs.

2 Signal detection in inverse problems
2.1 The minimax paradigm for inverse problems in signal detection
We consider the sequence space model (1)
yk = bk θk + εξk ,

k ∈ N∗ .

The signal detection problem aims at determining whether or not the observations y = (yk )k∈N∗
contains some signal. This question can be formalized as the following hypothesis testing
problem:
H0 : θ = 0l2 (N∗ ) against H1 : θ ∈ Θ, kθk2 ≥ rε2 ,
(3)
for some non decreasing positive sequence r = (rε )ε>0 depending on ε.
In the alternative hypothesis H1 , the set Θ denotes a subset of l2 (N∗ ). The requirement θ ∈ Θ

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can be thought either as a structural constraint on the signal or as a regularity condition on the
underlying function f in model (2). In the same time, the constraint kθk2 ≥ rε2 corresponds to
an energy condition that allows to quantify the amount of signal available in the observations.
Another problem closely related to signal detection is pattern recognition. In this case, one aims
at testing the adequation between the observations and a given reference signal θo . Having in
mind that up to the change of variable θ ← θ − θo , these two problems are equivalent we shall
only focus in the sequel on the signal detection problem.
In the sequel we denote as ∆ = ∆(y) a testing procedure. It is a measurable function of y =
(yk )k∈N∗ , such that ∆ ∈ {0, 1}, with the convention that we reject H0 if ∆ = 1 - that corresponds to
accept H1 - and do not reject the null hypothesis H0 otherwise. Given a level α ∈ (0, 1), ∆ = ∆α
is called a level-α test if and only if
P0l2 (N∗ ) (∆α = 1) ≤ α.
The risk under the alternative hypothesis is often measured through the maximal Type-II error
over the set Θ, as
β(∆α , Θ, rε ) := sup Pθ (∆α = 0).
θ∈Θ, kθk2 ≥rε2

In particular, given a level β ∈ (0, 1), the level-α test ∆α is said to be powerful if its maximal
Type-II error can be bounded by β, namely β(∆α , Θ, rε ) ≤ β. In this context, the minimax
paradigm has been at the core of several investigations over the last decades. Given both α and
β ∈ (0, 1) some fixed levels and a given set Θ, the separation rate Rε (∆α , Θ) associated to a given
level α-test ∆α is defined as
Rε (∆α , Θ) = inf {rε > 0, β(∆α , Θ, rε ) ≤ β} .
Typically, we expect that Rε (∆α , Θ) → 0 as ε → 0, although it strongly depends on the considered setting. The minimax separation rate associated to the testing problem (3) for a given set
Θ is then defined as r⋆ = (rε⋆ )ε>0 where
rε⋆ := inf Rε (∆, Θ),
∆

where the infimum is taken over all possible level-α tests ∆. We refer to Ingster and Suslina
(2003) or Baraud (2002) for exhaustive discussions on theses definitions. Determining the minimax rate of convergence for a given problem is quite informative. For substantial account on
the subject, see e.g., Ingster and Suslina (2003), Baraud (2002), Butucea (2007), Laurent et al.
(2012) or Lacour and Pham Ngoc (2014) among others.

2.2 Inverse and direct approaches
Several testing procedures have been proposed in order to deal with the testing problem (3). In
this section we will focus on two χ2 -based test statistics that have been proved to perform well

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in some minimax settings.
The amount of signal contained in the observations can be measured through the quantity kθk2 .
Then, a natural way to provide a decision rule is to construct an estimator of this quantity.
Inverse approach (IP): According to the sequence model (1), each coefficient θk can be estimated
∗
by b−1
k yk provided that the sequence of b = (bk )k∈N is known. For a given non decreasing
IP
sequence of non negative integers (Dε )ε>0 , this leads to the testing procedure ∆α,ε
IP
∆α,ε

= 1{T Dε >tα,ε } , where T Dε =

Dε
X
k=1

2
2
b−2
k (yk − ε )

(4)

and tα,ε is a threshold value that allows to control the Type-I error of the test. The integer Dε
plays a similar role than a regularization parameter in estimation. According to the choice of
the set Θ, specific choices for Dε are available. We refer, e.g., to Laurent et al. (2012) for more
details. We stress that a weighted variant of this procedure has been proposed and investigated
in several papers, as e.g. Ingster et al. (2012), allowing to obtain sharp asymptotic results.
Direct approach (DP): Since the sequence model (1) is derived from the model (2), we remark
that the test (4) is essentially based on an inversion of the operator A at hand. Such an inversion
appears to be quite natural in an estimation context in which we provide a reconstruction of
the unknown function f . In a signal detection framework, this is no longer required. Indeed,
setting ϑ = b.θ (i.e. ϑk = bk θk for all k ∈ N∗ ), we can remark that both assertion θ = 0l2 (N∗ ) and
ϑ = 0l2 (N∗ ) are equivalent. In other words, the testing problem, for some non decreasing positive
sequence µ = (µε )ε>0 :
H̃0 : ϑ = 0l2 (N∗ )

against

H̃1 : ϑ ∈ Θ̃, kϑk2 ≥ µ2ε ,

(5)

for some set Θ̃ only differs from (3) by the alternative. In some sense, (5) does not take into
account the fact that the data are distorted by a compact operator: we threat the data as a ‘direct’
problem and deal with a model of the form
Y = g + εξ

or

yk = ϑk + εξk

∀k ∈ N∗ ,

(6)

where g = A f ∈ K. Consequently, for a given non decreasing sequence of non negative integers
DP
(Dε )ε>0 we can introduce the test ∆α,ε
as
DP
∆α,ε

= 1{S Dε >sα,ε }

where

S Dε =

Dε
X
k=1

(y2k − ε2 ),

(7)

and S Dε corresponds to an estimator of kϑk2 and sα,ε denotes an appropriate threshold, allowing
a control of the Type-I error. This test provides interesting performance when dealing with (5).
But surprisingly, this is also the case for the testing problem (3) in some specific situations. We
refer to Laurent et al. (2011) for an extensive discussion on the subject with a minimax point of
view. One of the aim of this paper is to complete this discussion using a maxiset point of view.
This notion is extended to the signal detection context in the next section.

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2.3 The maxiset point of view in signal detection problem
In nonparametric function estimation, an estimator is a sequence (θ̂Dε )ε>0 possibly indexed by a
regularization (smoothing) parameter D and by the noise level ε, is a sequence (θ̂Dε )ε>0 . In the
minimax setting, given a functional set Θ, we determine sequence of rates r = (rε )ε>0 such that
for any ε ∈ (0, 1),
inf sup Eθ kθ̂ − θk2 ≥ crε2 and sup Eθ kθ̂Dε − θk2 ≤ Crε2 ,
θ̂

θ∈Θ

(8)

θ∈Θ

for some constant c, C > 0. Under the maxiset paradigm, we are giving a sequence of rates
r = (rε )ε>0 , and we exhibit the largest functional set Θ ⊂ l2 (N∗ ) for which (8) holds.
We will now adapt the maxiset point of view to the signal detection framework. Given a sequence r = (rε )ε>0 , we determine the largest set for which the maximal Type-II error can be
controlled with a prescribed error of our testing problem. This is formalized in the following
definition.
Definition 2.1 For a fixed (α, β) ∈ (0, 1/2)2 , let ∆α = (∆α,ε )ε>0 be a sequence of testing procedures and (rε )ε>0 a decreasing sequence of non negative real numbers. The maxiset of ∆α
associated to the separation rate r = (rε )ε>0 , is the largest sequence space Θ in l2 (N∗ ) such that,
for all ε ∈ (0, 1)
P0l2 (N∗ ) (∆α,ε = 1) ≤ α and

sup
θ∈Θ, kθk2 ≥rε2

Pθ (∆α,ε = 0) ≤ β.

This definition can be generalized in a straightforward way to the testing problem (5). In the
following, we denote the maxiset as MS (∆α , r) := MS ((∆α,ε )ε>0 , (rε )ε>0 ). Note that it clearly
corresponds to the following set:
io
h
n


MS (∆α , r) = (θk )k∈N∗ ∈ l2 (N∗ ) : ∀ε ∈ (0, 1), kθk2 ≥ rε2 ⇒ Pθ ∆α,ε = 0 ≤ β .

(9)

In Section 3, we shall derive an explicit expression of the maxisets for some tests based on χ2
statistics (see Theorem 3.1 and 3.2 below).
Following Definition 2.1 we let the reader be convinced that, for a given sequence of testing
procedures (∆α,ε )ε>0 , there is an embedding result between its maxisets associated with different
choices of detection rates.
Proposition 2.1 Let (rε′ ) and (rε ) two sequences of detection rates such that rε = o(rε′ ) as ε → 0.
Consider a sequence of testing procedures ∆α = (∆α,ε )ε>0 . Then, for some C > 0
MS (∆α , r) ⊂ MS (∆α , Cr′ ).
The previous embedding entails that the set of detectable functions increases as soon as we relax
the constraint on the lowest minimal energy required.

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3 Testing procedures and their maxiset performance
In this section, we first provide a description of the maxisets associated respectively to the
IP
DP
procedures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 defined in (4) and (7). From now on, we fix the two thresholds
tα,ε and sα,ε involved in the definition of these testing procedures as
v
tD
ε
X
p
2
2
tα,ε = Cα,1 ε
Dε ∀ε ∈ (0, 1),
(10)
b−4
and
s
=
C
ε
α,ε
α,2
k
k=1

where Cα,1 , Cα,2 > 0 denote two explicit constants that guarantee that the considered procedures
have a Type-I error controlled by α for all ε ∈ (0, 1). Undoubtedly, the smaller α is the bigger
Cα,1 and Cα,2 are. For the sake of simplicity, we do not use the 1 − α quantile of the respective
test statistics (S Dε )ε>0 and (T Dε )ε>0 . Such a change will not modify the spirit of the results displayed in this paper but will induce more technical details.
Below we characterize the maxisets associated to the considered procedure for general separation rates. In such a setting, these sets are poorly informative but they highlight valuable
information on the problem provided that we impose some structural constraints.

3.1 A general characterization of the maxisets
We start our investigations with a general description of the maxisets associated to the proceIP
DP
dures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 for any chosen rate of detection (rε )ε>0 . To this end, we introduce
the two sets Fr,D (C) and Gr,D (C) which will be of first importance in the sequel.
Definition 3.1 Let (Dε )ε>0 be an increasing sequence of non negative integers. Let (rε )ε>0 be a
decreasing sequence of non negative real numbers . For any C > 0 we set
v



tD

Dε


ε


X
X


 2



2
2
2
∗
−4


θ
>
Cε
,
kθk
≥
r
⇒
Fr,D (C) = 
b
θ
∈
l
(N
),
∀ε
∈
(0,
1);



2
ε
k
k









k=1
k=1



Dε

X
p






kθk2 ≥ r2 ⇒
∗
2 2
2

Gr,D (C) = 
D
θ
∈
l
(N
),
∀ε
∈
(0,
1);
b
θ
>
Cε
.

ε
2
ε

k
k




k=1

The following results emphasizes that these sets provide a characterization of the maxisets asIP
DP
sociated to the procedures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 .
Theorem 3.1 Consider (α, β) ∈ (0, 1/2)2 . Let (tα,ε )ε>0 and (sα,ε )ε>0 satisfying (10). Consider
IP
DP
the two sequences of testing procedures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 defined respectively in (4) and
(7). We have the two following maxiset results for any choice of detection rates r = (rε )ε>0 and
µ = (µε )ε>0 :

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1. There exist two positive constants Cmin (α, β) and Cmax (α, β) depending on Cα,1 and β such
that:
Fr,D (Cmax (α, β)) ⊂ MS (∆αIP , r) ⊂ Fr,D (Cmin (α, β)),
rewritten as: MS (∆αIP , r) = Fr,D .
′
′
2. There exist two positive constants Cmin
(α, β) and Cmax
(α, β) depending on Cα,2 and β such
that:
′
′
(α, β)),
(α, β)) ⊂ MS (∆αDP , µ) ⊂ Gµ,D (Cmin
Gµ,D (Cmax

rewritten as: MS (∆αDP, µ) = Gµ,D .
Remark 3.1 In Section 6, we provide explicit values of the constants Cmax (α, β) and Cmin (α, β)
′
′
(see (18) and (20)). The constants Cmax
(α, β) and Cmin
(α, β) are obtained from the values of
Cmax (α, β) and Cmin (α, β) by replacing Cα,1 by Cα,2 .
Surprisingly, the maxisets in the testing case have a completely different form compared to
results obtained in the estimation case. Indeed, according to Kerkyacharian and Picard (2002),
the constraint that a given procedure attains the rate (rε )ε>0 induces a tail constraint on the signal
of interest in the estimation problem. This is no more the case in the signal detection problem.
IP
DP
The Theorem 3.1 above indicates that the procedures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 are able to detect
2
2
only signals satisfying, for any ε ∈ (0, 1) such that kθk ≥ rε
v
tD
Dε
ε
X
X
θk2 > Cε2
b−4
∀ε ∈ (0, 1),
(11)
k ,
k=1

Dε
X
k=1

k=1

p
b2k θk2 > Cε2 Dε ,

∀ε ∈ (0, 1),

(12)

and for C large enough. In particular, the constraint (12) indicates that there should be enough
signal on the frequencies investigated by the test statistics T Dε . Nothing is said regarding the
high frequencies, i.e. the coefficient after the rank Dε . The maxiset results of Theorem 3.1,
constrasts with usual ones, since one does not describe maxisets in terms of smoothness spaces.
Moreover, this constraint has already been highlighted in, e.g. Laurent et al. (2012). Hence, the
maxiset paradigm is poorly informative in such a context. In the following section, we prove
that an additional structural assumption on the maxiset provides valuable informations on the
signal that can be detected by the procedures we are interested in.

3.2 A robust version of maxisets for tests
The main spirit of the previous section is that the functions that can be detected by the proceIP
DP
dures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 have enough energies for low frequencies. It means in particular
that our testing procedures are very sensitive to the trend of the signal. In what follows, we
shall require some robustness of our procedure with respect to this low frequency part of the

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signal, provided we have enough information. Indeed, in many practical situation, the signal
is preprocessed or filtered and we want to have theoretical guarantees about signal detection
remaining still valid in this context.
This structural constraint on the maxiset of interest, can be reformulated in a more formal way
as follows:
Definition 3.2 A set H ⊂ l2 (N∗ ) satisfies the decimation constraint if
θ∈H

⇒

θ(−n) ∈ H ∀n ∈ N∗ ,

where for any θ ∈ l2 (N∗ ) and n ∈ N∗ , θ(−n) = (θk 1{k>n} )k∈N∗ .
We stress that such a condition is for instance satisfied by all the sets Θ ⊂ l2 (N∗ ) of the form


+∞


X




γ
∗
Θ=
v
∈
l
(N
)
:
w
v
<
L
,

2
k k




k=1

for some positive sequence (wk )k∈N∗ and positive constants γ, L. Such a set describes some
smoothness conditions through the decay of the coefficients of the function of interest.
Hereafter we define two sequence spaces that are restriction of FD and GD to sequences satisfying the decimation constraint and that depend on the chosen detection rates r = (rε )ε>0 and
µ = (µε )ε>0 appearing in the definition of maxiset given in (9).
Definition 3.3 Let r = (rε )ε>0 be a decreasing sequence of non negative real numbers and
D = (Dε )ε>0 be an increasing sequence of non negative integers. For any C > 0 we set
v


tD


ε


X
X



∗
2
2
2
dec
−4 
,
θ
∈
l
(N
),
∀ε
∈
(0,
1);
θ
<
r
−
Cε
b
Fr,D (C) = 

2
ε
k
k






k>Dε
k=1
Gdec
r,D (C)




X
p 




∗
2 2
2
2
=
D
θ
∈
l
(N
),
∀ε
∈
(0,
1);
b
θ
<
r
−
Cε
.

ε
2
ε
k k




k>Dε

If we are searching for the largest set, satisfying both the requirement of Definition 2.1 and robust with respect to decimation, we define the so-called robust maxisets and we retrieve exactly
the sets introduced in the previous definition, up to some constants.
Undoubtedly, following the definition of D = (Dε )ε>0 , r = (rε )ε>0 and C the sequence spaces
above can be identical to the empty space. In the sequel, we especially focus on the cases where
these sequences spaces are not the empty space.
Definition 3.4 For any chosen sequences D = (Dε )ε>0 , r = (rε )ε>0 and C > 0, we say that
dec
(r, D, C) is F -admissible (respectively G-admissible) if and only if Fr,D
(C) (respectively Gdec
r,D (C))
is not the empty space.

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Remark 3.2 Following Definition 3.4, note that (r, D, C) is F -admissible and G-admissible for
rates of detection that do not converge to zero too fast as ε tends to zero.
Theorem 3.2 Consider (α, β) ∈ (0, 1/2)2 . Let (tα,ε )ε>0 and (sα,ε )ε>0 satisfying (10). Consider
IP
DP
the two sequences of testing procedures (∆α,ε
)ε>0 and (∆α,ε
)ε>0 defined respectively in (4) and
dec
IP
(7). We then respectively denote them by MS (∆α , r) and MS dec (∆αDP , µ) the respective robust
maxisets associated with the chosen rates r = (rε )ε>0 and µ = (µε )ε>0 . We have the following
maxiset results:
1. If (r, D, Cmax (α, β)) is F -admissible, then:
dec
Fr,D
(Cmax (α, β)) ⊂ MS dec (∆αIP , r) ⊂ F √dec
(Cmin (α, β)),
2r,D

(13)

dec
rewritten as: MS dec (∆αIP , r) = Fr,D
.
′
2. If (µ, D, Cmax
(α, β)) is G-admissible, then:
′
dec
′
√
Gdec
(∆αDP , µ) ⊂ Gdec
(Cmin
(α, β)),
µ,D (Cmax (α, β)) ⊂ MS
2µ,D

(14)

rewritten as: MS dec (∆αDP , µ) = Gdec
µ,D .
Remark 3.3 The constants stated in Theorem 3.2 are similar to those in Theorem 3.1.
We observe that, as in the estimation case, the maxiset with a decimation constraint depends
on the tail of the sequence (θk )k∈N∗ of interest. Note that in the framework of signal detection,
the situation is much more intricate than in estimation since one has several parameters to deal
with: the rate of convergence (rε )ε>0 , the nature of the operator eventually involved in the inverse
signal problem detection and β ∈ (0, 1) the Type-II error that has to be controlled.
According to the relative growth of the levels of possible energies of the signal and the sums of
the power of the eigenvalues of the operator involved in the signal detection problem, the nature
of the maxiset related to the sequence of testing procedures might be different. Consider the
IP
case where the sequence of testing procedures is ∆α,ε
. There are two extreme situations:
q
PDε −4
• First case: rε2 ≪ ε2
k=1 bk as ε → 0. In this case, Theorem 3.2 implies that the robust
maxiset is empty. It means that under the considered noise level ε, whatever the signal
we consider is, our procedure is never robust under decimation.
q
PDε −4
2
2
• Second case: rε ≫ ε
k=1 bk as ε → 0. In this case, the robust maxiset is non
empty and does not depend on the operator. In particular, provided that we have enough
energy in our signal, the performance of our detection procedure does not depend on the
underlying inverse problem we are considering.
q
PDε −4
The transition case where the two sequences rε2 and ε2
1 bk are equally balanced. Here,
the robust maxiset can be explicitly embedded as follows




X



∗
2
2
MS dec (∆αIP , r) ⊂ 
θ
∈
l
(N
),
∀ε
∈
(0,
1);
θ
<
r
,
2
ε
k




k>Dε

This version March 16, 2018

11

for some positive constant C. The set on the right-hand side of the previous embedding provides
a control on the tail of the sequence of interest by the considered rate (rε )ǫ>0 . In particular, the
faster the sequence (rε )ε>0 converges toward 0, the smoother the detectable function.
dec
Note that since the sequence space Fr,D
(C) may be empty or not, depending on the value of
IP
C, one cannot conclude that MS (∆α , r) is non-empty in whole generality. Similar comments
DP
are also valid if we considering the sequence of testing procedures ∆α,ε
and if we compare the
√
2
2
dec
behavior of the two sequences (µε ) and (ε Dε ) in the sequence space Fr,D
(C) as ε → 0.

4 Comparison of direct and inverse approaches
In this section, we will take advantage of tools developed in the previous section in order to
compare the direct and inverse approaches in a signal detection.
Indeed, we have seen that both problems (3) and (5) only differ by there alternatives. The tests
IP
DP
(∆α,ε
)ε>0 and (∆α,ε
)ε>0 have been specially designed in order to answer separately to each of
these problems. Now, a challenging question is to compare the alternative and to check whether
the inverse (resp. direct) approach is pertinent for the problem (5) (resp. (3)). This comparison
will be provided under the maxiset paradigm, using the robust version displayed in Section 3.2
above. To improve readability of our results, we now denote MS instead of MS dec , when denoting the robust maxisets.
In order to provide a fair comparison between both testing procedure, we have to state a dependency between the rates r = (rε )ε>0 and µ = (µε )ε > 0. Indeed, both procedures do not
come up in the same space. For instance, in the minimax paradigm, the rates are often faster
for ’direct’ alternative than for the inverse case. Below, we fix this dependency according to
previous calibration that has been investigated in the minimax paradigm (see, e.g. Laurent et al.
(2011)). Concerning the regularization parameter (Dε )ε>0 , we will keep the same value for both
testing procedure: the idea is to work with the same number of coefficients (same amount of
information).
Proposition 4.1 Fix (α, β) ∈ (0, 1/2)2 . Choose r = (rε )ε>0 , µ = (µε )ε>0 and D = (Dε )ε>0 such
that, for any ε ∈ (0, 1), µε = bDε rε . Then, provided that
v
u
t k
X
√
2
′
b−4
for all k ∈ N∗ ,
(15)
Cmax (α, β) k ≤ Cmin (α, β)bk
j
j=1

we get
MS (∆αIP , r) ⊂ MS (∆αDP , µ).

(16)

This version March 16, 2018

12

This proposition indicates that all the functions that can be detected by ∆αIP , can be also be
detected by ∆αDP . In other words, the direct test appears to be more efficient in the sense that
its associated maxiset is larger. One may ask the question of the strict inclusion. In order to
provide an answer, we will consider a specific setting and prove in particular that the inverse
testing procedure may miss some functions that can be detected by the direct case.
We are now considering the classical setting of the moderately ill-posed inverse problem, namely
we assume that for some t > 0 and any k, bk ∼ k−t . We also assume that we are
q in the case
PDε −4
where the calibration is the minimax one. In this case the two terms rε2 and
k=1 bk are
equally balanced, so that we are in the limit case described in Section 3.1.
4

Proposition 4.2 Let s, t > 0. Consider the case where for any ε ∈ (0, 1), Dε ∼ ε− 1+4(s+t) . Assume
4(s+t)
4s
that rε ∼ ε 1+4(s+t) and µε ∼ bDε rε ∼ ε 1+4(s+t) with b ∼ (bk )k∈N∗ ∼ (k−t )k∈N∗ . Then, there exist
functions θ such that
θ ∈ MS dec (∆αDP , µ)

but θ < MS dec (∆αIP , µ).

Remark 4.1 With the choice of operator given in Proposition 4.2, (15) is clearly satisfied and
therefore (16) holds.

5 Conclusion
In this paper, we adapt the maxiset approach for signal detection in inverse problems. This novel
tool for assessing the performance of testing procedures has been exposed to different classical
settings. In particular, it allows to compare the so-called direct and inverse approaches. We have
established that direct methods are associated to strictly larger maxisets in many cases, which
make such testing procedure more interesting for practical purpose.
This contribution provides a novel way for researchers to assess the performance of their testing
procedures. At the core of our future investigations will be adaptation of our methods to the
operator setting leading to the new concept of maxi-class.
In order to conclude this discussion, we mention that we are aware of a recent paper of Ermakov
(2018), that has been recently published in a similar setting while we were finalizing this article.
Although this paper also provide a definition of maxiset in testing context, we stress that it uses
different constraints on the set of interest. Moreover, it does not consider the inverse problem
setting, while we provide a comparison between direct and inverse approach. In our opinion,
both contributions are complementary and reveals different aspects of the same problem.

This version March 16, 2018

13

6 Proofs
6.1 Technical results
In this section, we recall and slightly extend some results that will be useful in the following.
More details regarding these results, e.g., context and extended discussions, can be found in
Baraud (2002) and Laurent et al. (2012).

Proposition 6.1 Fix (α, β) ∈ (0, 1)2 . There exists Cmax (α, β) and Cmin (α, β) such that, for all
ε ∈ (0, 1) and Dε ∈ N∗
v
tD
Dε
ε
X
X
IP
(i)
θk2 > Cmax (α, β)ε2
b−4
⇒ Pθ (∆α,ε
= 0) ≤ β,
k
(ii)

k=1

k=1

Dε
X

v
tD
ε
X

k=1

θk2 ≤ Cmin (α, β)ε2

b−4
k

k=1

IP
⇒ Pθ (∆α,ε
= 0) > β.

Proof: We start with the proof of item (i). A more precise proof is provided in Laurent et al.
(2012). In particular, the authors take advantage on available results on χ2 weighted statistics. This allows a better dependency of the constant Cmax (α, β) w.r.t. the values of α and β.
For the sake of completeness, we reproduce a simpler version of the proof, based on Markov
PDε −2 2
Inequality. Recall that we defined, for any ε ∈ (0, 1), T Dε as T Dε = k=1
bk (yk − ε2 ) and
q
PDε −4
tα,ε = Cα,1 ε2
k=1 bk for some constant Cα,1 > 0. Then
 P Dε 2
θk ,
• Eθ T Dε = k=1


PDε −4
PDε −2 2
bk + 4ε2 k=1
bk θ k .
• Varθ T Dε = 2ε4 k=1

Let θ = (θk )k∈N∗ be such that:

Dε
X
k=1

θk2 > Cmax (α, β)ε2

v
tD
ε
X
k=1

b−4
k .

(17)

This version March 16, 2018

14

Provided that Cmax (α, β) > Cα,1 , by using the Bienayme-Chebyshev inequality, one gets



IP
Pθ ∆α,ε
= 0 = Pθ T Dε ≤ tα,ε



= Pθ Eθ T Dε − T Dε ≥ Eθ T Dε − tα,ε

Varθ T Dε
≤


Eθ T Dε − tα,ε 2
PDε −4
PDε −2 2
2ε4 k=1
bk + 4ε2 k=1
bk θ k
= 
q
P Dε 2
PDε −4 2
2
θ
−
C
ε
α,1
k=1 k
k=1 bk
PDε −4
P Dε 2
2ε4 k=1
bk + 4ε2 maxl=1,...,Dε b−2
l ×
k=1 θk
≤
q

P
PDε −4 2
Dε 2
2
θ
−
C
ε
α,1
k=1 bk
k=1 k
2
4ε2 maxl=1,...,Dε b−2
2
l Cmax (α, β)
+
≤
P Dε 2
(Cmax (α, β) − Cα,1 )2
(Cmax (α, β) − Cα,1 )2 k=1
θk
2 + 4Cmax (α, β)
≤
(Cmax (α, β) − Cα,1 )2
≤ β.

The last inequality is obtained if Cmax (α, β) is large enough. More precisely, we can choose
Cmax (α, β) which depends on both Cα,1 and β as:
the maximum of Cα,1 and the positive solution of the following equation

2 + 4Cmax (α, β)
= β.(18)
(Cmax (α, β) − Cα,1 )2

Remark 6.1 We let the reader be convinced that the smaller β the larger the chosen Cmin (α, β)
and Cmax (α, β).


IP
We now prove the item (ii) of Proposition 6.1. To prove that Pθ ∆α,ε
= 0 > β is equivalent to
prove that

Pθ T Dε − E(T Dε ) ≥ tα,ε − E(T Dε ) < 1 − β.
(19)

To show this inequality, we apply Lemma 2 of Laurent et al. (2012) with x := − log(1 − β) and
4 P Dε
−4
2 P Dε
−2 2
σk ≡ εb−1
k . Setting Σ = ε
k=1 bk + 2ε
k=1 bk θk , we then get that (19) holds provided that
√
2
tα,ε − E(T Dε ) ≥ 2 Σx + 2 max b−2
l ε x.
l=1,...,Dε

Observe that
√
Σ ≤ ε2

v
tD
ε
X

≤ ε2

v
tD
ε
X

≤

b−4
k +

k=1

k=1

b−4
k +

v
t

2ε2

Dε
X

2
b−2
k θk

k=1

√
2ε max b−1
l
l=1,...,Dε



p
1 + 2Cmin (α, β) ε2

v
tD
ε
X
k=1

v
tD
ε
X
k=1

b−4
k ,

θk2

This version March 16, 2018

15

the last inequality comes from the assumption
Dε
X
k=1

θk2 ≤ Cmin (α, β)ε2

v
tD
ε
X

b−4
k .

k=1

√
Since β < 1/2, x < 1 (and therefore x < x). So, a sufficient condition to get the inequality
√
2
tα,ε − E(T Dε ) ≥ 2 Σx + 2 maxl=1,...,Dε b−2
l ε x is
Cα,1 ε2

v
tD
ε
X
k=1

v


tD
Dε
 p
 
ε
X
X

p
p

2
2
−4
−4 
ε
− log(1 − β).
bk − θk ≥ 2 1 + 2Cmin (α, β)
bk  ε2 − log(1 − β)+2b−2
Dε


k=1

k=1

This relationship is satisfied if Cmin (α, β) is small enough. The choice

! 12
q
p
p
Cmin (α, β) =
−2 log(1 − β) + (Cα,1 − 4 − log(1 − β)) − −2 log(1 − β)

(20)

IP
can be done in that context. This finishes the proof since this condition is compatible with ∆α,ε
is a α level test for a chosen Cα,1 sufficiently large enough.


Remark 6.2 The smaller β the more difficult the signal detection problem. This difficulty of the
signal detection problem is highlighted here in the value of Cmin (α, β). Indeed, we let the reader
be convinced that the smaller β the larger the chosen Cmin (α, β) and therefore the larger the l2
IP
norm of the the first terms of the signal θ have to be to ensure that Pθ (∆α,ε
= 0) ≤ β where
ε ∈ (0, 1).
In the same spirit of Proposition 6.1, the following result holds too.
′
′
Proposition 6.2 There exists Cmin
(α, β) and Cmax
(α, β) such that, for all ε ∈ (0, 1)

(i)
(ii)

Dε
X

p
′
b2k θk2 > Cmax
(α, β)ε2 Dε

DP
⇒ Pθ (∆α,ε
= 0) ≤ β,

k=1

p
′
b2k θk2 < Cmin
(α, β)ε2 Dε

DP
⇒ Pθ (∆α,ε
= 0) > β.

k=1
Dε
X

Since the proof of Proposition 6.2 is analogous to the one of Proposition 6.1, we omit it. We
′
′
precise that the values of Cmin
(α, β) and Cmax
(α, β) are obtained from the ones of Cmin (α, β) and
Cmax (α, β) by replacing Cα,1 by Cα,2 .


This version March 16, 2018

16

6.2 Proof of Theorem 3.1
We first prove the following embedding property : Fr,D (Cmax (α, β)) ⊂ MS (∆αIP , r) and thereafter
we prove that the embedding property MS (∆αIP , r) ⊂ Fr,D (Cmin (α, β)), whatever the choice of
the detection rate r = (rε )ε>0 .
First we prove that FD (Cmax (α, β)) ⊂ MS (∆αIP , r). Fix ε ∈ (0, 1) and let θ ∈ FD (Cmax (α, β))
satisfying kθk2 ≥ rε2 . Then
v
tD
Dε
ε
X
X
2
θk > Cmax (α, β)
b−4
k ⇒ Pθ (∆α,ε = 0) ≤ β.
k=1

k=1

This assertion is obvious thanks to item (i) of Proposition 6.1.
Conversely, we shall prove that MS (∆αIP , r) ⊂ FD (Cmin (α, β)), or equivalently that
θ < FD (Cmin (α, β)) ⇒ θ < MS (∆αIP , r).
Assume that θ < FD (Cmin (α, β)). Then there exists ε0 ∈ (0, 1) such that
v
u
t Dε
Dε0
X
X0
2
2
2
2
kθk ≥ rε0 and
θk ≤ Cmin (α, β)ε0
b−4
k .
k=1

k=1

Following item (ii) of Proposition 6.1, one deduces that
kθk2 ≥ rε20 and Pθ (∆α,ε0 = 0) > β.

(21)

Therefore we immediately deduce that θ < MS (∆αIP , r).
The proof of the second part of Theorem 3.1 follows exactly the same lines and is left to the
interested reader.

6.3 Proof of Theorem 3.2
As in the previous proof, we concentrate our attention on the maxiset associated to the seIP
quence of testing procedures (∆α,ε
)ε>0 . We recall that our aim is to prove that the robust maxiset
dec
IP
dec
MS (∆α , r) can be identified, up to a constant, to the set Fr,D
(C) defined as
v


tD


ε


X
X



∗
2
2
2
dec
−4 
θ
∈
l
(N
),
∀ε
∈
(0,
1);
θ
≤
r
−
Cε
,
Fr,D (C) = 
b
2
ε
k
k 






k>Dε
1

This version March 16, 2018

17

dec
We begin the proof with the first inclusion that is Fr,D
(Cmax (α, β)) ⊂ MS dec (∆αIP , r). It can be
easily seen that for all ε ∈ (0, 1)
v
tD
ε
X
X
dec
kθk2 ≥ rε2 and θ ∈ Fr,D
(Cmax (α, β)) ⇒
θk2 < kθk2 − Cmax (α, β)ε2
b−4
k
1

k>Dε

⇒

Dε
X

θk2 > Cmax (α, β)ε2

v
tD
ε
X

b−4
k

1

k=1

IP
⇒ Pθ (∆ε,α
= 0) ≤ β because of item (i) of Proposition 6.1.
dec
Since, one can easily check that Fr,D
(Cmax (α, β)) is stable by decimation, this entails that
dec
dec
IP
Fr,D (Cmax (α, β)) ⊂ MS (∆α , r).

Now, we turn our attention to the proof of the second inclusion that is MS dec (∆αIP , r) ⊂ F √dec (Cmin (α, β)).
2r,D
We consider the situation where for any ε ∈ (0, 1)
v
tD
ε
X
2
Cmax (α, β)ε2
b−4
k < rε
k=1

dec
(otherwise Fr,D
(Cmax (α, β)) is empty) Let θ < F √dec (Cmin (α, β)). In particular, since clearly
2r,D
Cmax (α, β) ≥ Cmin (α, β), there exists ε1 ∈ (0, 1) such that
v
v
u
u
t Dε
t Dε
X1
X1
X
2
2
2
b−4
≥
2r
−
C
(α,
β)ε
b−4
θk2 ≥ 2rε21 − Cmin (α, β)ε21
max
ε1
1
k
k > r ε1 .
k=1

k>Dε1

k=1

(−Dε1 )

Hence, the sequence (θk

)k∈N∗ satisfies both
X
IP
kθ(−Dε1 ) k2 =
= 0) > β,
θk2 > rε21 and Pθ(−Dε1 ) (∆α,ε
1
k>Dε1

where the last inequality is obtained thanks to item (ii) of Proposition 6.1 and the fact that
(−Dε )
θk 1 = 0 for all k ≤ Dε1 . This entails that θ(−Dε1 ) < MS (∆αIP , r) and therefore θ < MS dec (∆αIP , r)
because of the decimation constraint. Hence, we get
(Cmin (α, β)) ⇒ θ < MS (∆αIP , r)
θ < F √dec
2r,D
which implies that
(Cmin (α, β)).
MS dec (∆αIP , r) ⊂ F √dec
2r,D
This concludes the proof.


This version March 16, 2018

18

6.4 Proof of Proposition 4.1
The proof of Proposition 4.1 directly follows Theorem 3.2 and from the fact that
dec
′
Fr,D
(Cmin (α, β)) ⊂ Gdec
µ,D (Cmax (α, β)).
dec
Let us prove this embedding of sequence spaces. Assume first that θ ∈ Fr,D
(Cmin (α, β)), then
according to assumption (15) on the sequence (bk )k∈N∗ , for any ε ∈ (0, 1),
v


tD


ε
X
X

 X 

 .
θk2  < b2Dε rε2 − Cmin (α, β)ε2
b2k θk2 ≤ b2Dε 
b−4
k 


k>Dε

k>Dε

Since µε = bDε rε and b2Dε

k=1

q
′ (α,β) √
P Dε
Cmax
−4
4
Dε , we get
b
=
k=1 k
k=1 (b Dε /bk ) ≥ Cmin (α,β)
X
p
′
b2k θk2 < µ2ε − Cmax
(α, β)ε2 Dε

q
P Dε

k>Dε

′
which exactly means that θ ∈ Gdec
µ,D (Cmax (α, β)).

6.5 Proof of Proposition 4.2
We consider the case where for any k ∈ N∗ , bk ∼ k−t . One has
v
tD
ε
X
2 (1+4t)/2
rε2 ∼ ε2
b−4
∼ D−(1+4(s+t))/2
Dε(1+4t)/2 = Dε−2s .
ε
k ∼ ε Dε
k=1

We are then in the limit case. According to the value of C, Fr,D (C) may be empty or may be as
follows:

 





X
X








dec
2s
2
−2s
2
Fr,D
(C) := 
θ
:
∀ε
∈
(0,
1),
K
=
θ
:
sup
θ
≤
CD
θ
≤
C



ε
k
k





  K∈N∗

k>Dε

k>K

dec
which is a Besov space. Note that since the maxiset is only between the two sets Fr,D
(C1 ) and
dec
Fr,D (C2 ) with C1 and C2 unknown constants we cannot give an equality concerning the maxiset.
Using similar computations, we get that




X




2(s+t)
2 2
K
Gdec
(C)
=
θ
:
sup
b
θ
≤
C
.


µ,D
k
k


 K∈N∗

k>K

dec
′
Our aim below is to exhibit a sequence θ such that θ ∈ Gdec
µ,D (C) but θ < Fµ,D (C ), whatever the
value of the constant C ′ . To this end, let us consider the sequence θ = (θk )k∈N∗ such that

θk =

2− js
k

∀k ∈ {2 j , . . . , 2 j+1 − 1}, j ∈ N∗ .

Let K ∈ N∗ be fixed, and j0 such that 2 j0 ≤ K ≤ 2 j0 +1 . Then, we can check that
X
X
b2k θk2 .
k−(2t+2) 2−2 js . 2−2 j0 s K −2t ∼ K −2(s+t) .
k>K

k>K

This version March 16, 2018

19

In the same time, if K = 2 j0 ,
X
k>K

θk2

≥

0 +1 −1
2 jX

k=2 j0

2−2 js
∼ 2−2 j0 s ln(2 j0 +1 − 2 j0 ) ∼ 2−2 j0 s j0 >> K −2s ,
k2

when j increases.

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