Posterior Contraction and Credible Sets for Filaments of Regression
Functions

arXiv:1803.03898v1 [] 11 Mar 2018

Wei Li

∗

Subhashis Ghosal

†

North Carolina State University
March 13, 2018

Abstract
A filament consists of local maximizers of a smooth function f when moving in a certain direction.
Filamentary structures are important features of the shape of objects and are also considered as
important lower dimensional characterization of multivariate data. There have been some recent
theoretical studies of filaments in the nonparametric kernel density estimation context. This paper
supplements the current literature in two ways. First, we provide a Bayesian approach to the
filament estimation in regression context and study the posterior contraction rates using a finite
random series of B-splines basis. Compared with the kernel-estimation method, this has theoretical
advantage as the bias can be better controlled when the function is smoother, which allows obtaining
better rates. Assuming that f : R2 7→ R belongs to an isotropic Hölder class of order α ≥ 4, with
the optimal choice of smoothing parameters, the posterior contraction rates for the filament points
on some appropriately defined integral curves and for the Hausdorff distance of the filament are
both (n/ log n)(2−α)/(2(1+α)) . Secondly, we provide a way to construct a credible set with sufficient
frequentist coverage for the filaments. Our valid credible region consists of posterior filaments that
have frequentist interpretation. We demonstrate the success of our proposed method in simulations
and application to earthquake data.

1

Introduction

There is a large body of literature on the problem of estimating intrinsic lower dimensional structures
of multivariate data. A filament or a ridge line is one of such geometric objects that draws a lot of
attention in the recent years. Intuitively speaking, a filament consists of local maximizers of a smooth
function f (say a density or a regression function) when moving in a certain direction. Roughly speaking
these are generalized modes that reside on hyperplanes that are normal to the steepest ascent direction.
The filamentary structures (ridges) together with the valleys (i.e. local minimizers counterparts of
ridges), critical points are the main features of the shapes of objects. They are common in medical
∗
†

Department of Statistics, North Carolina State University. E-mail: wli35@ncsu.edu
Department of Statistics, North Carolina State University. E-mail: sghosal@stat.ncsu.edu

1

images, satellite images and many other three dimensional objects. One important example comes
from the study of the cosmic web — a large scale web structure of galaxies (clusters) connected by long
threads composed of sparse hydrogen gas. These intergalactic connections are believed to trace the
filaments of dark matters. The discovery and study of the dark matter is a key challenge of cosmology.
For more references, see Novikov, Colombi and Doré (2006), Dietrich, Werner, Clowe, Finoguenov,
Kitching, Miller and Simionescu (2012) and Chen, Ho, Brinkmann, Freeman, Genovese, Schneider and
Wasserman (2015).
The filament estimation falls into a broad category of data analytic methods that are called topological data analysis which is used to find intrinsic structure in data Wasserman (2016). In particular, it is
closely related to manifold learning problem. Manifold learning problem assumes that the data points
are generated from some a priori unknown lower dimensional structure with background noises. AriasCastro, Donoho and Huo (2006) developed a test to detect if a dataset contains some small fraction
of data points that are supported on a curve. Genovese, Perone-Pacifico, Verdinelli and Wasserman
(2012) and Genovese, Perone-Pacifico, Verdinelli and Wasserman (2014) studied the problem of estimation of the manifold. They showed that the ridge of a density function can serve as a surrogate to
the manifold and can be estimated with a better rate.
Recently, a promising algorithm called the Subspace Constrained Mean Shift (SCMS) algorithm,
is proposed in Ozertem and Erdogmus (2011). It can be considered as an algorithm that generalizes
the celebrated mean shift algorithm, which is designed for mode hunting Fukunaga and Hostetler
(1975). The consistency properties of SCMS has also been a subject of some recent research (see AriasCastro, Mason and Pelletier (2016)). As the filament is a feature induced by some unknown function,
say density, nonparametric estimation is often applied. The statistical properties of the estimated
filaments, like convergence rates, limiting distribution and the construction of valid confidence regions
are crucial for the assessment of the proposed method. There are a few recent papers on these issues.
Genovese et al. (2014) established the convergence rates of the filament obtained from the kernel density
estimation. Chen, Genovese and Wasserman (2015) provided Berry-Esseen type results for the limiting
distribution. With a different approach, Qiao and Polonik (2016) also established the convergence
rates and the extreme value type results for limiting distribution. Constructing confidence bands in
nonparametric problems is a very important topic in itself. Some recent papers include Claeskens
and Van Keilegom (2003), Giné and Nickl (2010) and Chernozhukov, Chetverikov and Kato (2013,
2014). Very little is known about how to construct confidence bands for a geometric object. In this
regard, Chen, Genovese and Wasserman (2015) developed a bootstrap-based method for uncertainty
quantification of filaments.
Frequentist properties of Bayesian procedure for an unknown function has also been an area receiving much attention. In particular, posterior contraction rates for functions in different Lr -metrics
have been studied in Giné and Nickl (2011); Castillo (2014) and Yoo and Ghosal (2016). Adaptive
contraction rates have been studied in De Jonge and Van Zanten (2012) for regression function and in
Shen and Ghosal (2015, 2016) for density function. The problem of constructing credible sets with adequate frequentist coverage has been studied in Knapik, van der Vaart and van Zanten (2011); Castillo
and Nickl (2013, 2014); Szabó, van der Vaart and van Zanten (2015); Ray (2017) and Yoo and Ghosal
2

(2016).
The filament estimation problem falls also into the category of problems where estimation and
inference is performed for a set, or more broadly speaking for some geometric object Molchanov (2006).
A prominent example is the level set of a certain function Jankowski and Stanberry (2012); Mason and
Polonik (2009); Mammen and Polonik (2013); Chen, Genovese and Wasserman (2017). Some other
examples arise in econometrics literature where partially identified parameters (as a set) is the object
of interest; see Chernozhukov, Hong and Tamer (2007); Moon and Schorfheide (2012) and references
therein. Very little is known about how to do inference about a geometric target in general and
this remains a very important area of study. Since the filaments can be considered as generalized
modes, relevant literature includes those on mode (and maximum) estimation, for instance, Shoung
and Zhang (2001); Facer and Müller (2003) and a recent paper by Yoo and Ghosal (2017) in the
Bayesian framework.
So far in the literature, the study of filaments is limited only to densities only using the kernel
approach. This paper supplements the current literature in two ways. First, we provide a Bayesian
approach to the filament estimation in regression context and study the contraction rates using a finite
random series of B-splines basis. This has theoretical advantages as the bias can be better controlled
when the function is more than “minimally smooth” (i.e. more than four times differentiable). Secondly,
we provide a way to construct credible set with sufficient frequentist coverage for the filaments. Different
from the bootstrap-based confidence region proposed by Chen, Genovese and Wasserman (2015) which
gives a band-shape region, our valid credible region consists of posterior filaments that have frequentist
interpretation. Another difference is that the inferential target in this paper is the true quantity itself,
while in Chen, Genovese and Wasserman (2015) the inference is targeted towards to the debiased
quantity.
Before we move on to the formal definition, it is worth to point out that some other possible
definitions of ridges or filaments have also been discussed and studied in mathematics and computer
sciences literature; see Eberly (1996) for more details. In this paper, we study the filament as introduced
in Eberly (1996), Chen, Genovese and Wasserman (2015), Qiao and Polonik (2016).
This paper is organized as follows. Notation and background materials are given in Section 2. The
model is formally introduced in Section 3 along with the prior distribution, and the posterior distribution is also described. Technical assumptions are given in Section 4. The main results in posterior
contraction and credible region are presented in Section 5. Simulation results and an application for
earthquake data are presented in Section 6 and 7 respectively. All proofs are given in Section 8.

2

Notations and preliminaries

Let N = {1, 2, .., }, N0 = {0, 1, 2, 3, . . .}. Given two real sequences an and bn , an = O(bn ) or an . bn
means that an /bn is bounded, while an = o(bn ) or an  bn means that both an /bn → 0. Also
an  bn means that an = O(bn ) and bn = O(an ). For random element Zn , Zn = Op (an ) means that
P(|Zn | ≤ Can ) → 1 for some constant C > 0.
P
For a vector x ∈ Rp , we define kxkp = ( pk=1 |xk |p )1/p for 0 ≤ p ≤ ∞, kxk∞ = max1≤k≤d |xk |
3

and write kxk for kxk2 . For a m × m matrix A, let kAk(r,s) = sup{kAxks : kxkr ≤ 1}. In particular,
P
kAk(2,2) = (λmax (AT A))1/2 , where λmax denotes the largest eigenvalue; kAk(∞,∞) = max1≤i≤m m
j=1 |aij |.
p
Let kAkF = tr(AT A) stands for the Frobenius norm of matrix A. We also denote an n by n identity
matrix by In .
For f : U 7→ R on some bounded set U ⊂ Rd . Let kf kp be the Lp norm and kf k∞ = supx∈U |f (x)|.
For g : U 7→ Rs on some bounded set U ⊂ Rd , let ∇g be the gradient of g, which is a s × d matrix
of functions. For a d-dimensional multindex r = (r1 , . . . , rd ) ∈ Nd0 , let Dr be the partial derivative
P
operator ∂ |r| /∂xr11 · · · ∂xrdd where |r| = dk=1 rk .
The Hölder Space Hα ([0, 1]d ) of order α > 0 consists of functions f : [0, 1]d 7→ R such that
kf kα,∞ < ∞, where k · kα,∞ is the Hölder norm
kf kα,∞ = max sup |Dr f (x)| + max
r:|r|≤bαc x

sup

r:|r|=bαc x,y:x6=y

|Dr f (x) − Dr f (y)|
,
kx − ykα−bαc

where sup is taken over the support of f and bαc is the largest integer strictly smaller than α.
The filament or the ridge line of a smooth function defined on R2 is a collection of points at which the
gradient of the function is orthogonal to the eigenvector of its Hessian whose corresponding eigenvalue
is negative. The filament point (that is the point on the filament) is a generalization of mode of the
function. To see this connection, recall a well-known result that tests for local maximum point (mode).
Let f : R2 7→ R be a smooth function, ∇f = (f (1,0) , f (0,1) )T be the gradient and Hf be the Hessian.
Recall a test for a local maximum point is the following
aT ∇f (x) = 0,

aT Hf (x)a < 0,

for all nonzero vector a. Let V (x) be the eigenvector of Hf (x) that corresponds to the smallest
eigenvalues λ(x). A point x ∈ R2 is called a ridge point if
V T (x)∇f (x) = 0,

V T (x)Hf (x)V (x) < 0.

Therefore, a ridge point is a point at which the function has a local maximum along the direction given
by V . Notice that V T (x)Hf (x)V (x) < 0 is equivalent to λ(x) < 0.
More generally, for f defined on Rd , 0 ≤ s ≤ d − 1, the eigenvectors of Hf (x) can be used to
define two orthogonal spaces, namely, a (d − s)-dimensional normal space (corresponding to (d − s)eigenvectors with smallest eigenvalues) and an s-dimensional tangent space (corresponding to the rest
eigenvectors). An s-dimensional ridge point on Rd is a point where the gradient of f is orthogonal
to the normal space and the eigenvalues associated with the normal space are all (strongly) negative.
Alternatively, such a point x can be regarded as a point where f attains the local maximum in the
affine space spanned by the normal space translated by x. The modes are then simply 0-dimensional
filaments. The 1-dimensional filament on R2 is of primary interest in our discussion here.
From now on, f is assumed to be some smooth regression function. Suppose that the Hessian matrix
Hf (x) of f at x has eigvenvector V (x) corresponding to the smaller eigenvalue λ(x). The filament
R(f ) of the regression function f is formally defined as
R(f ) = L := {x : h∇f (x), V (x)i = 0 and λ(x) < 0}.
4

We also introduce an integral curve, which is the solution of the following differential equation
dΥx0 (t)
= V (Υx0 (t)),
dt

Υx0 (0) = x0 .

Rt
Since Υx0 (−t) = 0 (−V (Υx0 (−s)))ds, with negative time it can be interpreted as a curve tracing in
the reverse direction, i.e, −V . Since the direction of V does not play a role in the theoretical study,
without loss of generality, t is restricted on the [0, Tmax ]. Define the “hitting time” of the filament by
the integral curve starting at point x0 ,
tx0 = argmin{t ≥ 0 : h∇f (Υx0 (t)), V (Υx0 (t))i = 0, λ(Υx0 (t)) < 0}.

3

Model, prior and posterior

Throughout the paper, let d = 2, thus x = (x1 , x2 ). We will use superscript * to emphasize the true
values. Assume that we have an independent and identically distributed sample (Yi , Xi )ni=1 from the
true distribution P0 ,
Yi = f ∗ (Xi ) + εi ,
where i are independent and identically distributed (i.i.d) sub-Gaussian random variable with mean
0 and variance σ02 for i = 1, . . . , n. Without loss of generality, we let Xi takes values in [0, 1]2 . Let
Y = (Y1 , . . . , Yn )T , X = (X1T , . . . , XnT )T , F ∗ = (f ∗ (X1 ), . . . , f ∗ (Xn ))T and ε = (ε1 , . . . , εn )T , then we
can write Y = F ∗ + ε. Our study allows both fixed and random design cases. If {Xi : i = 1, . . . , n} are
considered fixed data points, we assume that for some cumulative distribution function G with positive
and continuous density
 1 
sup |Gn − G| = o
,
(1)
J1 J2
x∈[0,1]2
i.i.d.

where Gn is the empirical distribution of {Xi : i = 1, . . . , n}. If Xi ∼ G, then the above condition is
satisfied with probability tending to one as long as J1  J2  o(n1/4 ) by Donsker’s theorem.
We shall use B-spline function as our basis function. For a sequence of knots 0 = t0 < . . . < tN +1 =
1, denote B-spline function of order q by Bi,q (x) = (ti − ti−q )[ti−q , ti ](· − x)q−1
+ , i = 1, 2, . . . , N + q
De Boor (1978). Here N is the number of interior points. By construction, Bi (x) > 0 on (ti−q , ti ) and
PN +q
2
i=1 Bi (x) = 1. Now to construct a basis on R , define bJ1 ,J2 ,q1 ,q2 (x) = (Bj1 ,q1 (x1 )Bj2 ,q2 (x2 ) : 1 ≤
jk ≤ Jk , k = 1, 2)T to be a vector of tensor product of B-splines functions, with possibly different orders
qj and knot sequences in different directions, i.e, 0 = tj,0 < tj,1 < . . . < tj,Nj < tj,Nj +1 = 1 for j = 1, 2.
Here Nj denotes the number of interior points and Jj = qj + Nj denotes the number of basis functions
on the jth coordinate. Whenever q1 , q2 are considered fixed, we shall suppress the subscripts q1 , q2 in
our notations of B-spline functions, for instance we write bJ1 ,J2 for bJ1 ,J2 ,q1 ,q2 . For each k = 1, 2, define
δk,` = tk,`+1 − tk,` for ` = 0, . . . , Nk . We assume that max1≤`≤Nk δk,` / min1≤`≤Nk δk,` ≤ C for some
C > 0. This assumption is clearly satisfied for the uniform partition.
Following a similar set-up in Yoo and Ghosal (2016), we first put a random tensor-product B series
prior on f . Let q1 , q2 fixed and Nj = Nj (n) and hence Jj = Jj (n), we model f (Xi ) = bTJ1 ,J2 (Xi )θ. Our
5

model becomes
Y |(X, θ, σ 2 ) ∼ N(Bθ, σ 2 In ),
where B = (bJ1 ,J2 (X1 ), . . . , bJ1 ,J2 (Xn ))T . We assign θ|σ 2 ∼ N(θ0 , σ 2 Λ0 ), assuming that for some
constants 0 < c1 ≤ c2 < ∞ we have that
c1 IJ1 J2 ≤ Λ0 ≤ c2 IJ1 J2 ,
where IJ1 J2 is a J1 J2 by J1 J2 identity matrix. It follows then

−1 −1
−1
T
−1
T
2
T
θ|Y, σ 2 ∼ N (Λ−1
.
0 + B B) (B Y + Λ0 θ0 ), σ (B B + Λ0 )
The posterior distribution for f (x) and its partial derivatives are then obtained accordingly. Define
the vector
!T
∂ r1
∂ r2
(r)
bJ1 ,J2 (x) =
Bj ,q (x1 ) r2 Bj2 ,q2 (x2 ) : 1 ≤ jk ≤ Jk , k = 1, 2 .
∂xr11 1 1
∂x2
Therefore,
Π(Dr f |Y, σ 2 ) ∼ GP(Ar Y + Cr θ0 , σ 2 Σr ),
(r)

T

(r)

T

−1 T
Ar (x) = bJ1 ,J2 (x) (B T B + Λ−1
0 ) B ,
−1 −1
Cr (x) = bJ1 ,J2 (x) (B T B + Λ−1
0 ) Λ0 ,
(r)

T

(r)

−1
Σr (x, y) = bJ1 ,J2 (x) (B T B + Λ−1
0 ) bJ1 ,J2 (y),

where we use GP to denote a Gaussian process. Notice that under P0 , Ar (x)ε/σ0 is a mean zero process
with sub-Gaussian tail.
To handle σ 2 , we can either put a conjugate inverse-gamma prior σ 2 ∼ IG(a/2, b/2) with shape
parameter a/2 > 2 and rate parameter b/2 > 0 or plug-in an estimate for σ 2 . Since the theory does
not make much difference, for ease of exposition, we shall use the second approach, to be called the
empirical Bayes method. The empirical Bayes has the following posterior distribution
Π(Dr f |Y, σ 2 )|σ2 =σ̂2 ∼ GP(Ar Y + Cr θ0 , σ̂ 2 Σr ),

(2)

where
σ̂n2 = n−1 (Y − Bθ0 )T (BΛ0 B T + In )−1 (Y − Bθ0 ).

4

Assumptions

We follow the standard assumptions in Qiao and Polonik (2016) . For convenience, we write Dr f = f (r)
and let d2 f (x) = (f (2,0) (x), f (1,1) (x), f (0,2) (x))T , ∇f = (f (1,0) , f (0,1) )T and Hf (x) be the Hessian of
f . We assume that the two distinct eigenvalues of Hf (x) are distinct. Then V (x) and λ(x) take the

6

following forms V (x) = G(d2 f (x)) and λ(x) = J(d2 f (x)) for some function G = (G1 , G2 )T : R3 → R2
and J : R3 → R given by
!
p
2u − 2w + 2v − 2 (w − u)2 + 4v 2
p
,
G(u, v, w) =
w − u + 4v − (w − u)2 + 4v 2 ,

p
1
J(u, v, w) =
u + w − (u − w)2 + 4v 2 .
2
Throughout the proofs, we may take the normalized version of the eigenvector V , that is, kV k = 1.
This is not necessary but it simplifies discussions.
We choose some compact subset G of [0, 1]2 so that t∗x0 is unique for any x0 ∈ G. Define L ⊕ δ =
∪x∈L B(x, δ), where B(x, δ) is an open ball around x of radius δ. The following assumptions will be
needed for the theory.
(A1) The truth f ∗ belongs to a Hölder Space Hα ([0, 1]2 ) with α ≥ 4.
(A2) There is some δ > 0 small, such that for all x ∈ L∗ ⊕ δ, Hf ∗ (x) has two distinct eigenvalues,
with smaller eigenvalue λ∗ (x) ≤ −η for some small positive value η.
(A3) For the δ > 0 in (A2), for all x ∈ L∗ ⊕ δ, |h∇h∇f ∗ (x), V ∗ (x)i, V ∗ (x)i| ≥ η for the same positive
value η in (A2).
(A4) The filament L∗ is a compact set such that L∗ = {Υ∗x0 (t∗x0 ) : x0 ∈ G}.
(A5) Assume that there exits a∗ > 0 such that for some CG > 0, for any x0 ∈ G,
inf

inf∗

x0 ∈G 0≤s<u<tx0 +a∗

Υ∗x0 (u) − Υ∗x0 (s)
≥ CG .
u−s

Assumption (A2) is important for our analysis as it guarantees the smoothness of V ∗ (ref. (3.1) of
Qiao and Polonik (2016) ) and that the normal vector of the filament ∇h∇f ∗ (x), V ∗ (x)i is well-defined.
Similar assumptions are needed in Genovese et al. (2014) and Chen, Genovese and Wasserman (2015).
Assumption (A3) says that the normal vector of the filament ∇h∇f ∗ (x), V ∗ (x)i is not orthogonal to
V ∗ (x). In addition, it implies that ∇h∇f ∗ (x), V ∗ (x)i =
6 0, i.e, the set {x : h∇f ∗ (x), V ∗ (x)i = 0} is not
“thick”. This means that a small change of x ∈ L necessarily changes the sign of h∇f ∗ (x), V ∗ (x)i. If one
restricts attention to the locus defined by h∇f ∗ (x), V ∗ (x)i = 0, noting rank(∇h∇f ∗ (x), V ∗ (x)i)=1, the
implicit function theorem says that L is a one-dimensional manifold in R2 . If for x ∈ L∗ , ∇f ∗ (x) = 0,
Assumption (A3) should be interpreted as |(V ∗ (x))T Hf ∗ (x)V ∗ (x)| > η which is then implied by
Assumption (A2). To see this, notice that
h∇h∇f ∗ (x), V ∗ (x)i, V ∗ (x)i = ∇f ∗ (x)T ∇V ∗ (x)V ∗ (x) + V ∗ (x)T Hf ∗ (x)V ∗ (x),
but V ∗ (x)T Hf ∗ (x)V ∗ (x) = λ∗ (x). Assumption (A3) parallels the assumption (A2) in Genovese et al.
(2014), where they assumed some upper bounds on the quantity related to the third derivative of the
density function; see also the assumption (P1) in Chen, Genovese and Wasserman (2015). Assumption
(A5) is common in the literature; see Koltchinskii, Sakhanenko and Cai (2007); Qiao and Polonik
(2016). Several useful consequences of these assumptions are summarized in Remark 8.1.
7

5

Posterior contraction and credible sets for filaments

In this section, we provide the main theoretical results. Here is a brief outline of these results. Theorem
5.1 provides posterior contraction rates for the integral curve. Proposition 5.2 gives posterior contraction rates for the hitting time. Theorem 5.3 gives the posterior contraction rates for the filament along
the integral curve. Theorem 5.8 establishes the posterior contraction rates for the filament around the
truth, posterior rates for deviations between the posterior filament and the filament induced by the
posterior mean, together with the convergence rates for the filament induced by posterior mean to that
of the truth, all in Hausdorff distance. Theorem 5.9 provides a valid credible set with sufficiently high
frequentist coverage.
The following result gives a Bayesian counterpart of Theorem 3.3 of Qiao and Polonik (2016). Our
proof is similar to theirs, but some technical details are different.
Theorem 5.1 Under Assumptions (A1),(A2) and (A5), for J1  J2  J  (n/ log n)1/(1+2α) , Tx∗0 =
t∗x0 + a∗ , we have the following posterior contraction rate, for n = (n/ log n)(2−α)/(1+2α) and any
Mn → ∞,
P0
Π( sup sup kΥx0 (t) − Υ∗x0 (t)k > Mn n |Y ) −→
0.
x0 ∈G t∈[0,Tx∗ ]
0

Remark 5.1 With the choice J  (n/ log n)1/(1+2α) , we obtain the same rate as in the Theorem 3.3 of
Qiao and Polonik (2016) up to a logarithmic term. For α = 4, the rate reduces to (n/ log n)−2/9 . Note
that if we choose J  (n/ log n)1/(2(1+α)) as it is the optimal choice for estimation of the function, it is
easy to see from the proof that now supx kd2 f (x) − d2 f ∗ (x)k2 is of order (n/ log n)(2−α)/(1+α) and the
term (3) in Section 8 has posterior contraction rate of order (n/ log n)(2−α)/(1+α) . Thus the contraction
rate will then be n = (n/ log n)(2−α)/(2(1+α)) , which is “suboptimal” in the present context.
The following proposition is a Bayesian analog of Proposition A.1 and Proposition 5.1 of Qiao and
Polonik (2016). We can obtain better rate by using different magnitude of the tuning parameter; see
the remark after the proposition.
Proposition 5.2 Under Assumptions (A1)–(A5), for J1  J2  J  (n/ log n)1/(1+2α) , we have the
following posterior contraction rate: for n = (n/ log n)(5−2α)/(2(1+2α)) and any Mn → ∞,
P

0
Π( sup |t∗x0 − tx0 | > Mn n |Y ) −→
0.

x0 ∈G

If in addition, ∇f ∗ (x) = 0 for all x ∈ L∗ , then the rates improve to n = (n/ log n)(2−α)/(1+2α) .
Remark 5.2 A better rate can be obtained if we choose J  (n/ log n)1/(2(1+α)) . With this choice,
by Lemma 8.2, supx k∇f (x) − ∇f ∗ (x)k has posterior contraction rate (n/ log n)(1−α)/(2(1+α)) , while
supx kV (x) − V ∗ (x)k has posterior contraction rate (n/ log n)(2−α)/(2(1+α)) . Recalling Remark 5.1, the
posterior contraction rate will then be n = (n/ log n)(2−α)/(2(1+α)) . If in addition, ∇f ∗ (x) = 0 for all
x ∈ L∗ , the function then is a plateau without any rise or fall along the direction of filament. In this
case, the rate improves to n = (n/ log n)(1−α)/(2(1+α)) .
8

Theorem 5.3 below is a Bayesian analog of the convergence of filaments points on the integral curves
starting from the same points; see Qiao and Polonik (2016), Section 3.4, for similar results. Again a
better rate is possible; see the remark following the theorem.
Theorem 5.3 Under Assumptions (A1)–(A5), for J1  J2  J  (n/ log n)1/(1+2α) , we have the
following posterior contraction rates: for n = (n/ log n)(5−2α)/(2(1+2α)) and any Mn → ∞,
P

0
0.
Π( sup kΥx0 (tx0 ) − Υ∗x0 (t∗x0 )k > Mn n |Y ) −→

x0 ∈G

Remark 5.3 A better rate can be obtained with the choice J  (n/ log n)1/(2(1+α)) , giving n =
(n/ log n)(2−α)/(2(1+α)) in view of Remark 5.1 and Remark 5.2.
In the following, we shall consider the Hausdorff distance between two filaments. Given two sets A
and B under Euclidean metric, let d(A|B) := supx∈A inf y∈B kx − yk. The Hausdorff distance between
A and B is defined as
Haus(A, B) = max{d(A|B), d(B|A)}.
In what follows, we provide a refined bound for the Hausdorff distance and construct credible sets for
the filaments. In fact, Theorem 5.3 gives an upper bound for the Hausdorff distance. However, for the
purpose of constructing credible sets with sufficient frequentist coverage, we need to have the upper
bound in terms of more primitive quantities such as the derivatives of underlying function. In view of
Remark 5.3, in this section we will restrict to the choice J  (n/ log n)1/(2(1+α)) .
Recall that f˜ = AY + Cθ0 is the posterior mean of f conditional on Y and that Ṽ ,Υ̃x0 , L̃ are the
corresponding eigenvector, integral curve and filament induced by f˜. In view of Lemma 8.3 and Lemma
8.4, following the proofs of Theorem 5.1, Proposition 5.2, Theorem 5.3, it is straightforward to show
that supx∈G supt kΥx0 (t) − Υ̃x0 (t)k, supx∈G ktx0 − t̃x0 k and supx∈G kΥx0 (tx0 ) − Υ̃x0 (t̃x0 )k are all small
with high posterior probability in P0 -probability. Likewise, it can be shown that the quantities induced
by f˜ converge to the corresponding true quantities induced by f ∗ . For instance Υ̃x0 (t) converges to
Υ∗x0 (t) uniformly in x0 ∈ G and t̃x0 converges to t∗x0 uniformly in x0 ∈ G and Υ̃x0 (t̃x0 ) converges to
Υ∗x0 (t∗x0 ) uniformly in x0 ∈ G in P0 -probability.
The following two theorems summarize above argument. Theorem 5.4 is on the convergence rates of
the Bayesian estimates of filaments to the true filaments. Theorem 5.5 is on the posterior contraction
rates of filaments around filaments induced by posterior mean.
Theorem 5.4 Under Assumptions (A1),(A2) and (A5), for J1  J2  J  (n/ log n)1/(2(1+α)) , Tx∗0 =
t∗x0 + a∗ , we have the following convergence rates:
sup

sup kΥ̃x0 (t) − Υ∗x0 (t)k = Op



 2−α 
n/ log n) 2(1+α) .

x0 ∈G t∈[0,Tx∗ ]
0

If in addition, (A3) and (A4) hold, then
sup |t̃x0 − t∗x0 | = Op



x0 ∈G

9

 2−α 
n/ log n) 2(1+α) ,

and
sup kΥ̃x0 (t̃x0 ) − Υ∗x0 (t∗x0 )k = Op



 2−α 
n/ log n) 2(1+α) .

x0 ∈G

Theorem 5.5 Under Assumptions (A1),(A2) and (A5), for J1  J2  J  (n/ log n)1/(2(1+α)) , Tx∗0 =
t∗x0 + a∗ , we have the following posterior contraction rates: for any Mn → ∞,
P

0
0.
sup kΥx0 (t) − Υ̃x0 (t)k > Mn (n/ log n)(3−2α)/(4(1+α)) |Y ) −→

Π( sup

x0 ∈G t∈[0,Tx∗ ]
0

If in addition, (A3) and (A4) hold, then
P

0
0,
Π( sup |tx0 − t̃x0 | > Mn (n/ log n)(2−α)/(2(1+α)) |Y ) −→

x0 ∈G

and
P

0
Π( sup kΥx0 (tx0 ) − Υ̃x0 (t̃x0 )k > Mn (n/ log n)(2−α)/(2(1+α)) |Y ) −→
0.

x0 ∈G

The following proposition says that with high posterior probability the induced filament from the
posterior satisfies similar properties the true filament has, so does the induced filament by f˜, with
P0 -probability tending to one.
Proposition 5.6 Suppose that f has a tensor-product B-splines prior with order q1 = q2 ≥ α and
J1  J2  (n/ log n)1/2(α+1) , then the following assertions hold. (i) The filament L of f drawn from
the posterior distribution satisfies assumptions (A2)–(A5) with posterior probability tending to 1 under
P0 -probability; (ii) The induced filament L̃ of the posterior mean f˜ satisfies assumptions (A2)–(A5)
with P0 -probability tending to 1.
Consider two filament curves L and L̂ that are close enough. The closedness can be guaranteed in
the probability limit argument in view of previous theorem if the inducing function f and fˆ are close
enough. Hence the second eigenvectors V and V̂ and the corresponding integral curves Υ and Υ̂ are
close enough. The following lemma is inspired by Genovese et al. (2014) Theorem 4, where they relate
the Hausdorff distance to kV k but under a different set of assumptions.
Lemma 5.7 Consider for two regression functions f and fˆ : [0, 1]2 7→ R that are sufficiently close
in supremum metric and both satisfy assumptions (A1)–(A4), then the Hausdorff distance between the
two induced filaments satisfies, for some positive constants C,
Haus(L, L̂) ≤


C
sup |f (2,0) (x)−fˆ(2,0) (x)|+ sup |f (1,1) (x)−fˆ(1,1) (x)|+ sup |f (0,2) (x)−fˆ(0,2) (x)| .
η x∈[0,1]2
x∈[0,1]2
x∈[0,1]2

So far, filaments are defined as a set on the plane. But one can also think about filaments that are
projected onto the surface of the functions, that is as a set in the three-dimensional space. Considering
the sets A = {(x, fˆ(x)) : x ∈ L̂}, B = {(x, f (x)) : x ∈ L}, we can obtain Haus(A, B)  Haus(L, L̂). To

10

see this, one first notices that the extra term supx∈L̂ inf y∈L |fˆ(x) − f (y)| shows up in the upper bound
of d(A|B). Let x0 ∈ L̂ as in the proof and notice that
inf |fˆ(x0 ) − f (y)| ≤ |fˆ(Υx0 (0)) − f (Υx0 (tx0 ))|

y∈L

≤ |fˆ(Υx0 (0)) − f (Υx0 (0))| + |f (Υx0 (0)) − f (Υx0 (tx0 ))|,

but then the last line is bounded by kfˆ − f k∞ + supx k∇f (x)k kΥx0 (tx0 ) − Υx0 (0)k. This indicates
d(A|B)  d(L̃|L) and hence the claim.
In view of Theorems 5.4, 5.5 and Proposition 5.6, we have the following result.
Theorem 5.8 Under Assumptions (A1)–(A5), with the choice of J1  J2  (n/ log n)1/(2(α+1)) , for
and Mn → ∞,
2−α

P

0
Π(Haus(L, L∗ ) > Mn (n/ log n) 2(α+1) |Y ) −→
0,

2−α

P

0
Π(Haus(L, L̃) > Mn (n/ log n) 2(α+1) |Y ) −→
0,

and
2−α 
Haus(L̃, L∗ ) = Op (n/ log n) 2(α+1) .

Remark 5.4 The third assertion of Theorem 5.8 for the convergence rate of the filament induced by
the posterior mean is an improvement over the rate in Theorem 5 of Genovese et al. (2014) when α > 4.
By the argument following Lemma 5.7, these results are directly applicable to the filaments projected
on the surface of the functions.
For the following result, we use k ∈ {1, 2, 3} instead of r as index for the order of derivatives,
that is f (1) = f (2,0) , f (1) = f (1,1) and f (1) = f (0,2) . Similarly, we define A(k) , C (k) , Σ(k) . Let f˜(k) :=
A(k) Y + C (k) θ0 be the posterior mean of f (k) and L̃ be the induced filament. For some 0 < γ < 1/2,
let Rn,k,γ denotes the 1 − γ quantile of the posterior distribution of kf (k) − f˜(k) k∞ . Let Cf,k,γ := {f :
kf (k) − f˜(k) k∞ ≤ ρRn,k,γ } for some large ρ > 0.
The following theorem provides two valid credible sets with sufficiently hight frequentist coverage
as the sample size increases. A similar method is used in Yoo and Ghosal (2017) where they provided
the credible sets for the mode and the maximum of a regression function.
Theorem 5.9 Assume (A1)–(A5), for the choice of J1  J2  (n/ log n)1/2(α+1) and any ρ > 1, for
the following two sets,
CL = {R(f ) : f ∈ ∩3k=1 Cf,k,γ },
C
C̄L = {L : Haus(L, L̃) ≤ ρ max Rn,k,γ },
η 1≤k≤3
the credibility and coverage probability of CL for L tend to 1 and CL ⊂ C̄L with high posterior probability
with P0 -probability tending to 1.
11

6

Simulation

Many algorithms have been proposed to find filaments. We here use the Subspace Constrained Mean
Shift (SCMS) algorithm proposed in Ozertem and Erdogmus (2011). The algorithm is also used in
Genovese et al. (2014), Chen, Genovese and Wasserman (2015) and Chen, Ho, Brinkmann, Freeman,
Genovese, Schneider and Wasserman (2015). Even though in the previous study, the algorithm is based
on kernel density estimator, our study suggests that nothing hinders the efficacy of the algorithm when
applied on a series based expansion in either regression framework or in density estimation. In the
following, we give a description of Subspace Constrained Mean Shift Algorithm.
Algorithm: (Subspace Constrained Mean Shift Algorithm)
Set  > 0, τ > 0, ā > 0 and select a collection of points {x1 , . . . , xn }, compute f (xi ) and keep only
(1)
those points for which f (xi ) > τ . Now for each xi , let xi = xi . Now iterate through the following
steps starting from t = 1:
(t)

(1) evaluate ∇f (xi );
(t)

(t)

(2) evaluate the Hessian Hf (xi ) and perform spectral decomposition to get V (xi ) the normalized
(t)
eigenvector of Hf (xi ) with the smallest eigenvalue;
(t+1)

(3) update xi

(t+1)

(4) stop if kxi

(t)

(t)

(t)

(t)

= āV (xi )V T (xi )∇f (xi ) + xi ;
(t)

(t)

(t)

− xi k <  or |V T (xi )∇f (xi )| < .
(t+1)

(t)

(t)

If in Step 3, one instead updates xi
= ā∇f (xi ) + xi , then it is just the celebrated mean shift
algorithm which can be used to find the mode of function f (the terminology “mean shift” can be better
(t)
understood when fˆ is estimated using a kernel density estimator, as then ∇f (xi ) is proportional to
the “mean-shift vector”).
In the simulation, we consider the following function
2
−1
 q

 1+cos (tan (x2 /x1 ))
2
2
f (x1 , x2 ) = 1 + φ x1 + x2

where φ(·) is the normal density function with mean 0.5 and standard deviation 0.3. We generate
i.i.d. data Xi uniformly on [0, 1] × [0, 1] and i.i.d. εi from normal with mean 0 and standard deviation
0.1 and then set Yi = f (Xi,1 , Xi,2 ) + i . The sample size is 2000. The Figure 1 shows function f
and its filament. The smoothing parameter (J1 , J2 ) plays an important role as in any nonparametric
estimation problem. We use fifth-order B-splines functions, that is q1 = q2 = 5. One can choose the
pair (J1 , J2 ) by their posterior mode (in a logarithmic scale) by maximizing the following
log Π(J1 , J2 |Y, σ̂ 2 ) = −2n log σ̂ − log(det(BΛ0 B T ) + In ) + const.

12

0.8

0.6

0.6

1.2

0.6

0.4

1.0

1.0

Figure 1: The function f and its filament.

0.4

0.6

0.8

1.2
1.6

1.6

0.6

0.5
X2

0.0

2.4

2.2

2.4

2.2

−0.5

0.0

2.2

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2

0.6

−0.5

X2

0.5

2

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0.8

2

1.8

0.8
1
0.8

−1.0

−0.5

0.0

0.5

1.4

−1.0

−1.0

1.8

1.4

0.6

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1.0

1

0.6

−1.0

0.0

X1

0.4

0.5

1.0

X1

0.2

0.4

1
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0.4

0

0.6

0.8

0

1.0

0.6

1.0

0.4

0.6

−0.5

1.4

1.8

2

0.5

2.4

2.4

1.8

X2

X2

0.0

2

2.2

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0.0

0.5

2.2
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X1

Figure 2: Effects of the smoothing parameter J1 and J2 . The orange circle is the truth, blue curve
(consists of dots) is estimated filament induced by posterior mean. Top left: J1 = J2 = 7; top right:
J1 = J2 = 8; bottom left: J1 = J2 = 9; bottom right: J1 = J2 = 15.
We use τ = 2, ā = 0.02 and  = 10−6 . Some pilot simulation suggests that (J1 , J2 ) = (9, 9) is
picked by its posterior mode throughout. Different choices have been experimented as well. In general,
“oversmoothing” may distort the filaments, whereas “undersmoothing” seems to produce similar results
as using J1 = J2 = 9, as can be seen in Figure 2. We also provide uncertainty quantification in Figure
3 with γ = 0.1, ρ = 1.2. Each graph shows 100 posterior filaments drawn from CL . To evaluate
13

Rn,k,γ , we first draw 200 posterior samples θ, compute their posterior mean θ̃. Next we compute
T

(k)

0.4

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1

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supx |bJ1 ,J2 (x) (θ − θ̃)| by searching on a crude grid and pick the maximum point on the grid and then
starting from this maximum point apply gradient ascent or descent method to check if nearby points
can achieve greater (absolute) value. We keep the largest value as the supremum. The (1−γ)-empirical
quantile over all these suprema will then be our Rn,k,γ . The posterior filaments that fall in the set CL
can then be generated.
To assess the performance over 100 iterations. The choice of ρ requires some pilot experiments.
It should not be too large in practice but gives a reasonable high percentage of f from posterior that
fall in Cf,k,γ . Some experiments suggests that ρ = 1.2 is a reasonable choice, giving 92.33% credibility
averaging over all iterations. To evaluate the coverage performance, we compute the Hausdorff distance
between L∗ and L̃. From the definition of C̄L , we set C/η to different values. Simulation shows that
L∗ belongs to C̄L for 91%, 94%, 98% time when C/η takes value 7.3 × 10−4 , 7.5 × 10−4 and 8 × 10−4
respectively. In general, C can be computed as supx kf˜(x)k and η as the smallest value of −λ along the
filament induced by the posterior mean. With this method, we obtain 100% coverage — high coverage
as the theory predicts.

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2.4

2.2

−0.5

0.6

2.2

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2.2

2.2

1.8

0.8

−1.0

−1.0

1.6
1.2

0.6

0.8

−1.0

−0.5

0.0

0.5

1.0

1.4

1

0.2

−1.0

X1

0.4

−0.5

0.8

0.0

0.4

0.5

1.0

X1

Figure 3: Uncertainty Quantification. Left: J1 = J2 = 7. Right: J1 = J2 = 9.

7

Application

For application, we use an earthquake dataset for California and its vicinity from January 1st of 2013
to December 31th of 2017 with magnitude 3.0 and above on the Richter scale 1 . The dataset consists of
3772 observations, among which 3383 observations have magnitude between 3 and 4; 355 observations
between 4 and 5; 34 observations above 5. The average magnitude is 3.439. The left panel in Figure 4
shows the data scatter plot. The sizes of circles are proportional to the magnitudes of the earthquakes.
In SCMS algorithm, we use ā = 5 × 10−6 and τ = 3 and  = 10−6 . We use q1 = q2 = 4 and
J1 = J2 = 32. We draw 200 posterior samples to compute the posterior mean. The filaments induced
by the posterior mean is plotted as the blue curve in Figure 4. The same filaments are overlayed on
1

The data s pub c y ava ab e rom https://earthquake.usgs.gov/earthquakes/

14

the magnitude surface as given in Figure 5. To obtain uncertainty quantification using filaments from
posterior samples, we use γ = 0.1, ρ = 1.2. The results showed that there are 91% of posteriors fall
into CL set, only slightly higher than the nominal credibility level. We randomly pick 100 of them to
plot the uncertainty quantification as given in the right panel of Figure 4.

Figure 4: Left: Earthquake data points (in red). Right: 100 filaments in the posterior constructed with
high frequentist coverage. The thick blue curves are the filaments induced by the posterior mean.

Figure 5: The magnitude surface and filaments induced by the posterior mean.
The filaments hence obtained provide useful characterization of the features of earthquake magnitude. Geographically, these filaments pass through the most populous coastal urban and suburban
areas in California, for instance, Eureka city, San Francisco and Los Angeles. Since this application
has utilized very small portion of earthquake data, we believe that a large scale study for different

15

periods of historical times will be useful for the study of the dynamics of the earthquakes. Uncertainty
quantification regions provide statistical understanding of what might be a reasonable shift of these
filaments through spatial and temporal domain and are also helpful for discovering newly emerging
crustal activities.

8

Technical Proofs

The posterior contraction results are implicitly assumed to be conditional on Y (for fixed design points)
or (Y, X) (for random design points) unless stated otherwise. In this section, we will provide lemmas
and formal proofs of the results stated in the main text. To focus on discussion, we will draw on several
useful results directly given in the following remark.
Remark 8.1 Under the assumptions (A1) to (A5), the following assertions hold.
(1) Integral curves are dense and non-overlapping.
(2) x0 7→ t∗x0 is continuous.
(3) kΥ∗x0 (t) − Υ∗x0 (t)k ≤ C exp(t)kx0 − x00 k.
0

(4) G = (G1 , G2 )T , ∇G are Lipschitz continuous and each element of Hessian HG1 , HG2 is bounded
on some open set Qδ ⊂ R3 such that {d2 f ∗ (x) : x ∈ [0, 1]2 } ⊂ Qδ . Thus V ∗ is Lipschitz
continuous.
There results can be proved using arguments similar to Qiao and Polonik (2016). Original argument
and proofs appear in that paper in various places. To save space, we do not provide details here, but
point to these following specific places in their paper. Result (1) is given in the discussion in page 10,
result (2) and (3) are discussed in pages 22 and 48, and result (4) is discussed in pages 53–55.

8.1

Some lemmas

Lemma 8.1 For f ∈ Hα ([0, 1]2 ) with α ≤ min(q1 , q2 ). Then for J1  J2 sufficiently large, there exists
a function f ∞ := bTJ1 ,J2 θ∞ for some θ∞ such that
kf

∞

− f k∞ ≤ C

2
X

Jk−α ,

k=1

kDr f ∞ − Dr f k∞ ≤ C

2
X

Jkrk −α ,

k=1

for some positive constant C depending on α and q and for every integer vector r satisfying |r| < α.
Proof. For proof, see Schumaker (2007).

16

Lemma 8.2 (Posterior contraction around truth) Under above assumptions, with J1  J2  J and
J 2 ≤ n, chosen such that δn,k,J vanishes to zero, then for any Mn → ∞,
P

0
Π( sup |Dr f (x) − Dr f ∗ (x)| > Mn δn,|r|,J |Y ) −→
0,

x∈[0,1]2

P

0
Π( sup k∇f (x) − ∇f ∗ (x)k > Mn δn,1,J |Y ) −→
0,

x∈[0,1]2
P

0
Π( sup kd2 f (x) − d2 f ∗ (x)k > Mn δn,2,J |Y ) −→
0,

x∈[0,1]2
P

0
Π( sup kHf (x) − Hf ∗ (x)kF > Mn δn,2,J |Y ) −→
0,

x∈[0,1]2
P

0
0,
Π( sup k∇d2 f (x) − ∇d2 f ∗ (x)kF > Mn δn,3,J |Y ) −→

x∈[0,1]2
P

0
Π( sup kV (x) − V ∗ (x)kF > Mn δn,2,J |Y ) −→
0,

x∈[0,1]2
P

0
Π( sup k∇V (x) − ∇V ∗ (x)kF > Mn δn,3,J |Y ) −→
0.

x∈[0,1]2

where δn,k,J = J k (log n/n)J 2 + J −2α

1/2

.

Proof. These results can be directly adapted from Yoo and Ghosal (2016) by noting the highest degree
of derivatives in each expression. We shall show the rates for supx kHf (x)−Hf ∗ (x)kF , supx k∇d2 f (x)−
∇d2 f ∗ (x)kF and supx k∇V (x) − ∇V ∗ (x)kF . Notice that
1/2

,
kHf (x) − Hf ∗ (x)kF = (f (2,0) (x) − f ∗(2,0) (x))2 + 2(f (1,1) (x) − f ∗(1,1) (x))2 + (f (0,2) (x) − f ∗(0,2) (x))2

≤ 4−1/2 |f (2,0) (x) − f ∗(2,0) (x)| + 2|f (1,1) (x) − f ∗(1,1) (x)| + |f (0,2) (x) − f ∗(0,2) (x)| .
The rate for supx kHf (x) − Hf ∗ (x)kF then follows easily.
Also, the contraction rates for k∇d2 f (x) − ∇d2 f ∗ (x)kF follows from the inequality k∇d2 f (x) −
P
∇d2 f ∗ (x)kF ≤ 6−1/2 r:|r|=3 |Dr f (x) − Dr f ∗ (x)|.
Lastly, since ∇V (x) − ∇V ∗ (x) = ∇G(d2 f (x))∇d2 f (x) − ∇G(d2 f ∗ (x))∇d2 f ∗ (x), which is 2 by 2
matrix. Straightforward calculation gives its (1, 1) element


(1,0,0) 2
(1,0,0) 2 ∗
G1
(d f (x))f (3,0) (x) − G1
(d f (x))f ∗(3,0) (x) ,


(0,1,0) 2
(0,1,0) 2 ∗
+ G1
(d f (x))f (2,1) (x) − G1
(d f (x))f ∗(2,1) (x) ,


(0,0,1) 2
(0,0,1) 2 ∗
+ G1
(d f (x))f (1,2) (x) − G1
(d f (x))f ∗(1,2) (x) .
The absolute value of the first summand is bounded by the sum of
(1,0,0)

G1

(1,0,0)

(d2 f (x))f ∗(3,0) (x) ,

(1,0,0)

(d2 f ∗ (x))f ∗(3,0) (x) .

(d2 f (x))f (3,0) (x) − G1

and
(1,0,0)

G1

(d2 f (x))f ∗(3,0) (x) − G1
17

Noting that d2 f (x) contracts to d2 f ∗ (x) uniformly in x, hence {d2 f (x) : x ∈ [0, 1]2 } ⊂ Qδ with posterior probability tending to 1, the first term is bounded by a constant multiple of |f (3,0) (x) − f ∗(3,0) (x)|
, in view of the result 4 of Remark 8.1. By the same remark and assumption kf ∗ kα,∞ < ∞, the
second term is bounded by kd2 f (x) − d2 f ∗ (x)k|f ∗(3,0) (x)|. Using similar arguments for the second
and third summand, one can see that the absolute value of the (1,1) element of ∇V (x) − ∇V ∗ (x) is
bounded by |f (3,0) (x) − f ∗(3,0) (x)| + |f (2,1) (x) − f ∗(2,1) (x)| + |f (1,2) (x) − f ∗(1,2) (x)|. Dealing the rest
P
elements of ∇V (x)−∇V ∗ (x) similarly, we can see that k∇V (x)−∇V ∗ (x)kF . r:|r|=3 |Dr f −Dr f ∗ |.
In above lemma, the optimal rates are obtained when J  (n/ log n)1/2(α+1) , which then yields
δn,k,J = n  (log n/n)(α−k)/(2α+2) .
In addition, we have the following two lemmas whose proofs closely follow that of Lemma 8.2 and
thus are omitted. Denote the posterior mean of f by f˜ := AY + Cθ0 and similarly define the quantities
it induces, for instance, Ṽ ,Υ̃x0 and L̃. We then have the following lemma.
Lemma 8.3 (Posterior contraction around the posterior mean) Under above assumptions and J1 
J2  J and J 2 ≤ n, with J is chosen such that ηn,k,J vanishes to zero, then for any Mn → ∞,
P

0
Π( sup |Dr f (x) − Dr f˜(x)| > Mn ηn,|r|,J |Y ) −→
0,

x∈[0,1]2

P0
Π( sup k∇f (x) − ∇f˜(x)k > Mn ηn,1,J |Y ) −→
0,
x∈[0,1]2
P

0
Π( sup kd2 f (x) − d2 f˜(x)k > Mn ηn,2,J |Y ) −→
0,

x∈[0,1]2
P0
Π( sup kHf (x) − H f˜(x)kF > Mn ηn,2,J |Y ) −→
0,
x∈[0,1]2
P

0
Π( sup k∇d2 f (x) − ∇d2 f˜(x)kF > Mn ηn,3,J |Y ) −→
0,

x∈[0,1]2
P

0
Π( sup kV (x) − Ṽ (x)kF > Mn ηn,2,J |Y ) −→
0,

x∈[0,1]2
P

0
Π( sup k∇V (x) − ∇Ṽ (x)kF > Mn ηn,3,J |Y ) −→
0,

x∈[0,1]2

where ηn,k,J = J k+1 (log n/n)1/2 .
One can also readily obtain the following convergence rates for the Bayesian estimators induced by
f˜.
Lemma 8.4 (Convergence of posterior mean) Under the above assumptions and J1  J2  J and

18

J 2 ≤ n, with J is chosen such that δn,k,J vanishes to zero, then we have
sup |Dr f˜(x) − Dr f ∗ (x)| = Op (δn,|r|,J ),

x∈[0,1]2

sup k∇f˜(x) − ∇f ∗ (x)k = Op (δn,1,J ),
x∈[0,1]2

sup kd2 f˜(x) − d2 f ∗ (x)k = Op (δn,2,J ),
x∈[0,1]2

sup kH f˜(x) − Hf ∗ (x)kF = Op (δn,2,J ),
x∈[0,1]2

sup k∇d2 f˜(x) − ∇d2 f ∗ (x)kF = Op (δn,3,J ),
x∈[0,1]2

sup kṼ (x) − V ∗ (x)kF = Op (δn,2,J ),
x∈[0,1]2

sup k∇Ṽ (x) − ∇V ∗ (x)kF = Op (δn,3,J ),
x∈[0,1]2

where δn,k,J = J k (log n/n)J 2 + J −2α

1/2

.

Lemma 8.5 If the Condition (1) holds, then we have
!
2
Y
Jk−1 IJ1 J2 ≤ B T B ≤ C2 n
C1 n

2
Y

k=1

!
Jk−1

IJ1 J2 ,

k=1

for some constants C1 , C2 > 0. If in addition, for some constants 0 < c1 ≤ c2 < ∞ such that
c1 IJ1 J2 ≤ Λ ≤ c2 IJ1 J2 ,
we have
C1 n

2
Y

!
Jk−1

+

c−1
2

T

IJ1 J2 ≤ B B + Λ

−1

≤

C2 n

k=1

2
Y

!
Jk−1

+

c−1
1

IJ1 J2 .

k=1

Proof. See Yoo and Ghosal (2016) Lemma A.9. and discussion in p.1075.

8.2

Proofs of the main results

.
Proof of Theorem 5.1. We first sketch a proof for the following result as first proved in Koltchinskii
et al. (2007):
Z t
∗
sup sup
Υx0 (t) − Υx0 (t) . sup sup
(V − V ∗ )(Υ∗x0 (s))ds .
x0 ∈G t∈[0,Tx∗ ]
0

x0 ∈G t∈[0,Tx∗ ]
0

0

To see this, let
Z t

yx0 (t) := Υx0 (t) − Υ∗x0 (t) =
V (Υx0 (s)) − V ∗ (Υ∗x0 (s) ds,
0
Z t
Z t
∗
∗
zx0 (t) :=
(V − V ) Υx0 (s)ds +
∇V ∗ (Υ∗x0 (s))zx0 (s)ds.
0

0

19

Rt
Note that δx0 (t) := yx0 (t) − zx0 (t) = 0 ∇V ∗ (Υ∗x0 (s))δ(s)ds + Rx0 (t) for some reminder term Rx0 (t). So
Rt
kδx0 (t)k ≤ kRx0 (t)k + 0 k∇V ∗ (Υ∗x0 (s))kF kδx0 (s)kds, by the Gronwall-Bellman inequality Kim (2009),


Z t
∗
∗
k∇V (Υx0 (s))kF ds .
kRx0 (t)k exp
kδx0 (t)k ≤ sup
t∈[0,Tx∗0 ]

0

Hence supt∈[0,Tx∗ ] kδx0 (t)k) . supt∈[0,Tx∗ ] kRx0 (t)k. It can be showed that with high posterior prob0
0
R Tx∗0
ability supt∈[0,Tx∗ ] kRx0 (t)k  0 kyx0 (s)kds in P0 -probability tending to 1 following the argument
0
R Tx∗
in page 1586 of Koltchinskii et al. (2007). Therefore, supt∈[0,Tx∗ ] kδx0 (t)k  0 0 kyx0 (s)kds. Since
R T0x∗
yx0 (t) = δx0 (t) + zx0 (t),it follows that supt∈[0,Tx∗ ] kδx0 (t)k  0 0 kzx0 (s)kds . supt∈[0,Tx∗ ] kzx0 (t)k.
0
0
Then supt∈[0,Tx∗ ] kyx0 (t)k = supt∈[0,Tx∗ ] kδx0 (t) + zx0 (t)k . supt∈[0,Tx∗ ] kzx0 (t)k. But
0

0

0

Z

t

kzx0 (t)k ≤
0

(V − V ∗ ) Υ∗x0 (s)ds +

Z
0

t

k∇V ∗ (Υ∗x0 (s))kF kzx0 (s)kds.

One more application of the Gronwall-Bellman inequality and then take the supremum on the left hand
side yields
 Z t
Z t

∗
∗
∗
∗
sup kzx0 (t)k ≤ sup
(V − V ) Υx0 (s)ds exp
k∇V (Υx0 (s))kF ds
t∈[0,Tx∗0 ]

t∈[0,Tx∗0 ]

0

Z
.

sup
t∈[0,Tx∗0 ]

0

t

0

(V − V ∗ ) Υ∗x0 (s)ds .

Taking another supremum over all x0 gives us the result.
Now coming back to the main proof, by an integral form Taylor expansion of V (x)(= G(d2 f (x)))
around G(d2 f ∗ (x)), by the uniform boundedness of each element of second derivative G, one can get
Z t


sup
(V − V ∗ )(Υ∗x0 (s)) − ∇G(d2 f ∗ (Υ∗x0 (s)))d2 (f − f ∗ )(Υ∗x0 (s)) ds . sup kd2 f (x)−d2 f ∗ (x)k2 .

x0 ∈G
t∈[0,Tx∗0 ]

x0

0

In view of Lemma 8.2, with the choice of J in the assumption, supx kd2 f (x) − d2 f ∗ (x)k2 is of order
(n/ log n)(5−α)/(1+2α) . Now it suffices to prove that
Z t
sup
∇G(d2 f ∗ (Υ∗x0 (s)))d2 (f − f ∗ )(Υ∗x0 (s))ds
x0 ∈G
t∈[0,Tx∗0 ]

0

has posterior contraction rate (n/ log n)(2−α)/(1+2α) . To this end, we demonstrate the following quantity
has this posterior contraction rate
Z t
∂G1 2 ∗ ∗
∗
sup
(d f (Υx0 (s)))(f (2,0) − f (2,0) )(Υ∗x0 (s))ds .
x0 ∈G
0 ∂x1
t∈[0,Tx∗0 ]

The rate for the other components can be derived similarly. Let Un be a shrinking neighborhood of σ02
such that σ̂n2 ∈ Un and Π(σ 2 ∈ Un |Y ) → 1 with probability 1.
20

∂G1 2 ∗
∗
∂x1 (d f (Υx0 (s))).

Let G̃(Υ∗x0 (s)) =

To derive the rate, it suffices to show the rate for

Z t
h

∗
E0 sup E sup
G̃(Υ∗x0 (s))(f (2,0) − f (2,0) )(Υ∗x0 (s))ds
x0 ,t

σ 2 ∈Un

2

Y, σ 2

i

(3)

0

decays as (n/ log n)(4−2α)/(1+2α) , which then by Markov’s inequality yields the desired posterior contraction rate. Notice that the posterior variance of f (2,0) (x) does not depend on Y , while its posterior
mean A(2,0) Y + C(2,0) θ0 does not depend on σ 2 . Also, in the posterior distribution conditional on σ 2 ,
(2,0) T

−1/2 Z, where Z is a J 2 × 1 vector
we have f (2,0) (x) − A(2,0) (x)Y − C(2,0) (x)θ0 = σbJ1 ,J2 (x)(B T B + Λ−1
0 )
of standard normal random variables. Thus we can write
Z t
h

i
2
∗
E0 sup E sup
G̃(Υ∗x0 (s))(f (2,0) − f (2,0) )(Υ∗x0 (s))ds Y, σ 2
x0 ,t

σ 2 ∈Un

. E0

h

0

Z t

sup E sup
G̃(Υ∗x0 (s))(f (2,0) − A(2,0) Y − C(2,0) θ0 )(Υ∗x0 (s))ds
x0 ,t

σ 2 ∈Un

2

σ2

i

0

Z t
h
2i
∗
+ E0 sup
G̃(Υ∗x0 (s))(A(2,0) Y + C(2,0) θ0 − f (2,0) )(Υ∗x0 (s))ds
x0 ,t

0

2

Z



. sup σ E0 sup
x0 ,t

σ 2 ∈Un

Z

t

+ sup
x0 ,t

0

0

t

(2,0) T

−1/2
G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))ds(B T B + Λ−1
Z
0 )

G̃(Υ∗x0 (s))(A(2,0) (Υ∗x0 (s))F ∗

+

C(2,0) (Υ∗x0 (s))θ0

−f

(2,0) ∗

2

2

(Υ∗x0 (s)))ds



Z t
2
∗
∗
+ E0 sup
G̃(Υx0 (s))A(2,0) (Υx0 (s))εds
.
x0 ,t

(4)
(5)
(6)

0

Rt
(2,0)
We shall tackle each of above three terms one by one. Let a(x0 , t) = 0 G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))ds,
where the integral is taken elementwise. Now consider the first term (4). Define U1,n (x0 , t) =
−1/2 Z which is Gaussian. For fixed x , t,
a(x0 , t)(B T B + Λ−1
0
0 )
E[U1,n (x0 , t)]2 = a(x0 , t)T (B T B + Λ−1
0 )a(x0 , t) .

J5
J2
ka(x0 , t)k2 .
,
n
n

where last step follows from Lemma 8.6. Consider the two points (x0 , t) and (x̃0 , t̃), where the distance
q
d((x0 , t), (x̃0 , t̃)) := Var(U1,n (x0 , t) − U1,n (x̃0 , t̃))

1/2
. (a(x0 , t) − a(x̃0 , t̃))T (B T B + Λ−1
)(a(x
,
t)
−
a(x̃
,
t̃))
0
0
0
 J 2 1/2
.
ka(x0 , t) − a(x̃0 , t̃)k
n
J7
1/2
.
(|t − t̃| + kx0 − x̃0 k2 )
;
n
˜ 0 , t), (x̃0 , t̃)) := (|t − t̃| + kx0 − x̃0 k2 )1/2 . Thus
here the last line follows from Lemma 8.6. Let d((x
˜ 0 , t), (x̃0 , t̃)) for ρn = J 7/2 /n1/2 . By Lemma A.11 of Yoo and Ghosal (2016),
d((x0 , t), (x̃0 , t̃)) . ρn d((x


setting δn  1/J there, we obtain E supx0 ,t [U1,n (x0 , t)]2 = O(J 5 log n/n). Therefore, (4) is of order
(log n)J 5 /n.
21

For the term (6), let U2,n (x0 , t) =
Observe that for fix x0 , t,

Rt
0

−1 T
G̃(Υ∗x0 (s))A(2,0) (Υ∗x0 (s))εds = a(x0 , t)(B T B + Λ−1
0 ) B ε.

−1 −1
−1 T
T
E[U2,n (x0 , t)]2 = σ02 a(x0 , t)T (B T B + Λ−1
0 ) B B(B B + Λ0 ) a(x0 , t)

J2 n J2
a(x0 , t)T a(x0 , t)
n J2 n
J5
. σ02 .
n
. σ02

Since ε is sub-Gaussian,
U2,n (x0 , t) is sub-Gaussian
with mean 0 and variance E[U2,n (x0 , t)]2 . By the


same argument, E supx0 ,t [U2,n (x0 , t)]2 = O(J 5 log n/n). Therefore, (6) is of order (log n)J 5 /n.
Finally, for (5), write
Z

t

0

Z

0

+

C(2,0) (Υ∗x0 (s))θ0

−f

(2,0) ∗



(Υ∗x0 (s))

ds



(2,0) T
−1
T
∗
−1
G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))(B T B + Λ−1
(θ
−
θ
)
B
(F
−
Bθ
)
+
Λ
)
0
∞ ds
∞
0
0

t



∗
(2,0) T
G̃(Υ∗x0 (s)) bJ1 ,J2 (Υ∗x0 (s))θ∞ − f (2,0) (Υ∗x0 (s)) ds .

0

+

A(2,0) (Υ∗x0 (s))F ∗

t

=
Z



G̃(Υ∗x0 (s))

The first term on the right hand side of above equation is just


−1
−1
T
∗
(θ
−
θ
)
,
)
B
(F
−
Bθ
)
+
Λ
a(x0 , t)T (B T B + Λ−1
0
∞
∞
0
0
which is bounded by


−1
−1
T
∗
ka(x0 , t)k1 k(B T B + Λ−1
)
k
kB
(F
−
Bθ
)k
+
kΛ
k
(kθ
k
+
kθ
k
)
.
∞
∞
0
∞
∞
∞
(∞,∞)
(∞,∞)
0
0
−1
2
T
∗
Note that ka(x0 , t)k1 . J 2 by Lemma 8.6. Since k(B T B + Λ−1
0 ) k(∞,∞) = O(J /n) and kB (F −
Bθ∞ )k∞ = O(nJ −2−α ), this term is of order (J 4 /n) + J 2−α . The second term of the right hand side
of above equation can be bounded by the supremum of approximate error which is of order J 2−α .
Therefore, (5) is of order J 4 ((J 4 /n2 ) + J −2α ).
Putting all the above terms together, (3) is of order (log n/n)J 5 + J 4 ((J 4 /n2 ) + J −2α ). For
J  (n/ log n)1/(1+2α) , (3) is of order (n/ log n)(4−2α)/(1+2α) , which then completes the proof.

Lemma 8.6 For any x0 , x̃0 ∈ G, t ∈ [0, Tx∗0 ] and t̃ ∈ [0, Tx̃∗0 ], let r = (2, 0), (1, 1) or (0, 2), a(x0 , t) :=
Rt
(r)
∗
∗
0 G̃(Υx0 (s))bJ1 ,J2 (Υx0 (s))ds, where the integral is taken elementwise, under Assumption (A1), (A2)
and (A5),
ka(x0 , t)k1 . J 2 ,
ka(x0 , t)k2 . J 3 ,
ka(x0 , t) − a(x̃0 , t̃)k2 . J 3 |t − t̃| + J 5 kx0 − x̃0 k2 .

22

Proof. We shall consider only two separate cases (i) r = (2, 0) and (ii) r = (1, 1).
(i). For r = (2, 0) (similarly for r = (0, 2)),
ka(x0 , t)k1 =

J1 X
J2 Z
X
j1 =1 j2 =1

.

0

J1 X
J2 Z
X
j1 =1 j2 =1 0

≤

J1 Z
X
j1

the last line follows by

PJ2

j2 =1 Bj2 (x2 )

0

t

t

t

G̃1 (s)Bj001 (Υ∗1,x0 (s))Bj2 (Υ∗2,x0 (s))ds
|Bj001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))ds

|Bj001 (Υ∗1,x0 (s))|ds,

= 1 and noticing that

(q1 − 1)(q1 − 2)Bj1 ,q1 −2 (x)
(q1 − 1)(q1 − 2)Bj1 −1,q1 −2 (x)
+
(tj1 − tj1 −q1 +2 )(tj1 − tj1 −q1 +1 ) (tj1 −1 − tj1 −q1 +1 )(tj1 − tj1 −q1 +1 )
(q1 − 1)(q1 − 2)Bj1 −2,q1 −2 (x)
(q1 − 1)(q1 − 2)Bj1 −1,q1 −2 (x)
+
,
−
(tj1 −1 − tj1 −1−q1 +2 )(tj1 −1 − tj1 −q1 ) (tj1 −2 − tj1 −1−q1 +1 )(tj1 −1 − tj1 −q1 )

Bj001 ,q1 (x) =

P
which implies Jj11=1 |Bj001 (Υ∗1,x0 (s))| . 4J 2 . Therefore, ka(x0 , t)k1 . J 2 .
Next, let Sj1 = [t1,j1 −(q1 −2) , t1,j1 ], Sj2 = [t2,j2 −q2 , t2,j2 ] and 1j1 ,j2 (s) := 1{s : Υ∗x0 (s) ∈ Sj1 × Sj2 }.
Turn to ka(x0 , t)k2 , which is
J2 Z
J1 X
X
j1 =1 j2 =1

Z tZ
=
0

.

0

G̃1 (s)G̃1 (s0 )

J1 X
J2 
X


Bj001 (Υ∗1,x0 (s))Bj2 (Υ∗2,x0 (s))Bj001 (Υ∗1,x0 (s0 ))Bj2 (Υ∗2,x0 (s0 )) dsds0

j1 =1 j2 =1
J1 X
J2 
t X

0 j =1 j =1
1
2

Z tZ
=

0

2
G̃1 (s)Bj001 (Υ∗1,x0 (s))Bj2 (Υ∗2,x0 (s))ds

0

Z tZ
0

t

t

J1 X
J2
t X

0 j =1 j =1
1
2


|Bj001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))|Bj001 (Υ∗1,x0 (s0 ))|Bj2 (Υ∗2,x0 (s0 )) dsds0



1j1 ,j2 (s)1j1 ,j2 (s0 ) |Bj001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))|Bj001 (Υ∗1,x0 (s0 ))|Bj2 (Υ∗2,x0 (s0 )) dsds0 .

The last equality is obtained as follows. Since for any fix j1 , j2 , Bj001 (·) is supported on Sj1 and Bj2 (·)
is supported on Sj2 . So 1j1 ,j2 (s)1j1 ,j2 (s0 ) = 1{(s, s0 ) : Υ∗x0 (s) ∈ Sj1 × Sj2 and Υ∗x0 (s0 ) ∈ Sj1 × Sj2 }.
Notice that for n large, |Sj1 |  |Sj2 |  J −1 and Υ∗x0 (s) ∈ Sj1 × Sj2 and Υ∗x0 (s0 ) ∈ Sj1 × Sj2 implies that
kΥ∗x0 (s)−Υ∗x0 (s0 )k . J −1 . By Assumption (A5), CG |s−s0 | ≤ |s−s0 k(Υ∗x0 (s)−Υ∗x0 (s0 ))/(s−s0 )k . J −1 ,
and hence |s − s0 | . J −1 . Therefore, for n large enough, above quantity can be further bounded by a
constant multiple of
Z tZ
0

J1 X
J2
t X

0 j =1 j =1
1
2





1{|s − s0 | < CJ −1 } |Bj001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))|Bj001 (Υ∗1,x0 (s0 ))|Bj2 (Υ∗2,x0 (s0 )) dsds0 .

23

PJ2

Noting Bj2 (x2 ) ≤ 1 and
Z tZ
0

.

J1 X
J2
t X

0 j =1 j =1
1
2

Z tZ tX
J1
0

0

. J 2 J −1

j2 =1 Bj2

= 1, it can be further bounded by




1{|s − s0 | < CJ −1 } |Bj001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))|Bj001 (Υ∗1,x0 (s0 ))| dsds0




1{|s − s0 | < CJ −1 } |Bj001 (Υ∗1,x0 (s))Bj001 (Υ∗1,x0 (s0 ))| dsds0

j1

Z

J1
t X

0 j =1
1

|Bj001 (Υ∗1,x0 (s))|ds.

From argument used in bounding ka(x0 , t)k1 , we have
proof for ka(x0 , t)k2 . J 3 .

PJ1

00
∗
j1 =1 |Bj1 (Υ1,x0 (s))|

. J 2 . This completes the

For the third result, we write
ka(x0 , t) − a(x̃0 , t̃)k2 . ka1 (x0 , t, t̃)k2 + ka2 (t̃, x0 , x̃0 )k2 ,
where
t

Z

(2,0)

G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))ds −

a1 (x0 , t, t̃) :=
0

t̃

Z
0

(2,0)

G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))ds,

and
t̃

Z
a2 (t̃, x0 , x̃0 ) :=
0

(2,0)
G̃(Υ∗x0 (s))bJ1 ,J2 (Υ∗x0 (s))ds

Z
−
0

t̃

(2,0)

G̃(Υ∗x̃0 (s))bJ1 ,J2 (Υ∗x̃0 (s))ds.

First, note that
2

ka1 (x0 , t, t̃)k =

J1 X
J2 Z
X
t̃

j1 =1 j2 =1

. J 2 J −1

Z

t

2

G̃(Υ∗x0 (s))Bj001 (Υ∗1,x0 (s))Bj2 (Υ∗2,x0 (s))ds

t̃

(4J 2 )ds

t
3

= J |t − t̃|,
where the second line follows by a similar argument used to bound ka(x0 , t)k2 . Next, ka2 (t̃, x0 , x̃0 )k2
is given by
J1 X
J2 hZ t̃ 
i2
X
G̃(Υ∗x0 (s))Bj001 (Υ∗1,x0 (s))Bj2 (Υ∗2,x0 (s)) − G̃(Υ∗x̃0 (s))Bj001 (Υ∗1,x̃0 (s))Bj2 (Υ∗2,x̃0 (s))ds
,
j1 =1 j2 =1

0

which is bounded (up to a constant multiple ) by
J1 X
J2 Z t̃


2
X
G̃(Υ∗x0 (s)) Bj001 (Υ∗1,x0 (s)) − Bj001 (Υ∗1,x̃0 (s)) Bj2 (Υ∗2,x0 (s))ds
j1 =1 j2 =1

+

J1 X
J 2 Z
X
j1 =1 j2 =1

+

0

0

t̃


2

G̃(Υ∗x0 (s)) Bj2 (Υ∗2,x0 (s)) − Bj2 (Υ∗2,x̃0 (s)) Bj001 (Υ∗1,x̃0 (s))ds

J1 X
J2 Z t̃ 
X
j1 =1 j2 =1

0


2
G̃(Υ∗x0 (s)) − G̃(Υ∗x̃0 (s)) Bj001 (Υ∗1,x̃0 (s))Bj2 (Υ∗2,x̃0 (s))ds .

24

Bound the first term in the right hand side of above expression as
J1 X
J 2 Z
X

2


G̃(Υ∗x0 (s)) Bj001 (Υ∗1,x0 (s)) − Bj001 (Υ∗1,x̃0 (s)) Bj2 (Υ∗2,x0 (s))ds

t̃

2
|G̃(Υ∗x0 (s))||Bj0001 (Υ∗1,x0 (s))||Υ∗1,x0 (s) − Υ∗1,x̃0 (s)|Bj2 (Υ∗2,x0 (s))ds

0

j1 =1 j2 =1

J1 X
J2  Z
X

.

t̃

0

j1 =1 j2 =1

.kx0 − x̃0 k

J1 X
J2  Z
X

2

0

j1 =1 j2 =1

.kx0 − x̃0 k2 J 2

Z

t̃

0

t̃

2
|G̃(Υ∗x0 (s))||Bj0001 (Υ∗1,x0 (s))|Bj2 (Υ∗2,x0 (s))ds

|Bj0001 (Υ∗1,x0 (s))|ds

2 5

.kx0 − x̃0 k J .
The second term is bounded as
J1 X
J 2 Z
X

J1 X
J 2 Z
X

t̃

2
|G̃(Υ∗x0 (s))||Υ∗2,x0 (s) − Υ∗2,x̃0 (s)||Bj0 2 (Υ∗2,x̃0 (s))||Bj001 (Υ∗1,x̃0 (s))|

0

j1 =1 j2 =1

.kx − x̃0 k



2
G̃(Υ∗x0 (s)) Bj2 (Υ∗2,x0 (s)) − Bj2 (Υ∗2,x̃0 (s)) Bj001 (Υ∗1,x̃0 (s))ds

0

j1 =1 j2 =1

.

t̃

2

J1 X
J2  Z
X
j1 =1 j2 =1

.kx − x̃0 k2

t̃

Z
0

×

J1 X
J2
X

t̃

0

2
|G̃(Υ∗x0 (s))||Bj0 2 (Υ∗2,x̃0 (s))||Bj001 (Υ∗1,x̃0 (s))|

1{|s − s0 | ≤ CJ −1 }

|Bj0 2 (Υ∗2,x̃0 (s))||Bj001 (Υ∗1,x̃0 (s))||Bj0 2 (Υ∗2,x̃0 (s0 ))||Bj001 (Υ∗1,x̃0 (s0 ))|dsds0

j1 =1 j2 =1

.kx − x̃0 k

2

t̃

Z
0

5

1{|s − s | ≤ CJ }J
0

−1

2

J1
X

|Bj001 (Υ∗1,x̃0 (s))||Bj001 (Υ∗1,x̃0 (s0 ))|dsds0

j1 =1

2

.J kx0 − x̃0 k ,
where the second line follows from the mean value theorem, the third line from the Lipschitz continuity
of Υ∗x0 in x0 (Remark 8.1) whereas fourth line follows by a similar argument used to bound ka(x, t)k2 .
The third term
J1 X
J2 Z t̃ 
X
j1 =1 j2 =1

.

0


2
G̃(Υ∗x0 (s)) − G̃(Υ∗x̃0 (s)) Bj001 (Υ∗1,x̃0 (s))Bj2 (Υ∗2,x̃0 (s))ds

J1 X
J2 Z t̃ 
X
j1 =1 j2 =1 0

2
kΥ∗x0 (s) − Υ∗x̃0 (s)kBj001 (Υ∗1,x̃0 (s))Bj2 (Υ∗2,x̃0 (s))ds

25

.kx0 − x̃0 k

2

J1 X
J 2 Z
X

t̃

0

j1 =1 j2 =1

2
|Bj001 (Υ∗1,x̃0 (s))|Bj2 (Υ∗2,x̃0 (s))ds

.kx0 − x̃0 k2 J 3 ,
where the second line holds by mean value theorem and the third line holds due to Lipschitz continuity
of Υ∗x0 in x0 (Lemma 8.1) and last line holds by similar argument for ka(x0 , t)k2 .
In summary, we have that ka2 (t̃, x0 , x̃0 )k2 . J 5 kx0 − x̃0 k2 and ka1 (x0 , t, t̃)k2 . J 3 |t − t̃|.
(ii). Now turning to the case r = (1, 1). By similar argument we have
J1 X
J2 Z t
X
ka(x0 , t)k1 =
G̃1 (s)Bj0 1 (Υ∗1,x0 (s))Bj0 2 (Υ∗2,x0 (s))ds
0

j1 =1 j2 =1

.

J1 X
J2 Z
X
j1 =1 j2 =1 0
J1 Z
X

≤J

t

j1 =1 0

Likewise, ka(x0 , t)k2 =
Z tZ
0

J1 X
J2
t X

0 j =1 j =1
1
2

.J
0

.J

2

0

.J

3

J1
t X

0 j =1
1

Z tZ
0

. J 3 J −1

t

0

Z

|Bj0 1 (Υ∗1,x0 (s))|ds . J 2

2
0 (Υ∗ (s))B 0 (Υ∗ (s))ds
G̃
(s)B
can be bounded by
1
1,x0
2,x0
j1
j2
0

PJ2 R t
j2 =1



0 j =1 j =1
1
2

Z tZ

j1 =1

|Bj0 1 (Υ∗1,x0 (s))kBj0 2 (Υ∗2,x0 (s))|ds



1{|s − s0 | < CJ −1 } |Bj0 1 (Υ∗1,x0 (s))Bj0 2 (Υ∗2,x0 (s))Bj0 1 (Υ∗1,x0 (s0 ))Bj0 2 (Υ∗2,x0 (s0 ))| dsds0

J1 X
J2
t X

Z tZ

PJ1

t





1{|s − s0 | < CJ −1 } |Bj0 1 (Υ∗1,x0 (s))kBj0 2 (Υ∗2,x0 (s))kBj0 1 (Υ∗1,x0 (s0 ))| dsds0




1{|s − s0 | < CJ −1 } |Bj0 1 (Υ∗1,x0 (s))Bj0 1 (Υ∗1,x0 (s0 ))| dsds0

1{|s − s | < CJ }
0

−1

J1 
X


|Bj0 1 (Υ∗1,x0 (s))| dsds0

j1
J1
t X

0 j =1
1

|Bj0 1 (Υ∗1,x0 (s))|ds

. J 3.
The third result ka1 (x0 , t, t̃)k2 . J 3 |t − t̃| and ka2 (t̃, x, x̃0 )k2 . J 5 kx0 − x̃0 k2 can be derived in a similar
manner, since
2
J1 X
J2 Z t
X
ka1 (x0 , t, t̃)k2 =
G̃(Υ∗x0 (s))Bj0 1 (Υ∗1,x0 (s))Bj0 2 (Υ∗2,x0 (s))ds
t̃

j1 =1 j2 =1

. J 2 J −1

Z

t̃

(4J 2 )ds

t

= J 3 |t − t̃|,
26

where the second line follows by a similar argument used in bounding ka(x0 , t)k2 . Next, to bound
ka2 (t̃, x0 , x̃0 )k2 , we need to estimate the following three terms
J1 X
J 2 Z
X
j1 =1 j2 =1



2
G̃(Υ∗x0 (s)) Bj0 1 (Υ∗1,x0 (s)) − Bj0 1 (Υ∗1,x̃0 (s)) Bj0 2 (Υ∗2,x0 (s))ds ,

t̃



2
G̃(Υ∗x0 (s)) Bj0 2 (Υ∗2,x0 (s)) − Bj0 2 (Υ∗2,x̃0 (s)) Bj0 1 (Υ∗1,x̃0 (s))ds ,

0

J1 X
J 2 Z
X
j1 =1 j2 =1

t̃

0

J1 X
J2 Z t̃ 
X
j1 =1 j2 =1

0


2
G̃(Υ∗x0 (s)) − G̃(Υ∗x̃0 (s)) Bj0 1 (Υ∗1,x̃0 (s))Bj0 2 (Υ∗2,x̃0 (s))ds .

This can be done by similar argument used for the case r = (2, 0) and we omit the details.
Proof of Proposition 5.2 . We will first sketch a proof for that for any  > 0,
P

0
Π( sup |t∗x0 − tx0 | > |Y ) −→
0.

x0 ∈G

t∗x0

Recall that
= argmin{t ≥ 0 : h∇f ∗ (Υ∗x0 (t)), V ∗ (Υ∗x0 (t))i = 0, λ∗ (Υ∗x0 (t)) < 0} Let Cx0 = {t ∈
[0, t∗x0 − ] : h∇f ∗ (Υ∗x0 (t)), V ∗ (Υ∗x0 (t))i = 0}. Noticing that the existence of unique t∗ is equivalent to
the following set of assertions,
1. inf x0 ∈G,t∈Cx0 λ∗ (Υ∗x0 (t)) > 0,
2. inf x0 ∈G,t∈[0,tx0 −]\Cx0 |∇f ∗ (Υ∗x0 (t)), V ∗ (Υ∗x0 (t))i| > 0,
3. supx0 ∈G,t∈[t∗x

0

−,t∗x0 +] λ

∗ (Υ∗ (t))
x0

< 0,

4. For all x0 ∈ G, there exists tx0 ∈ [t∗x0 − , t∗x0 + ] such that
∇f ∗ (Υ∗x0 (tx0 )), V ∗ (Υ∗x0 (tx0 ))i = 0.
By invoking the arguments similar to that in the proof of Proposition A.1 of Qiao and Polonik (2016),
with P0 -probability tending to 1, the above assertions hold with high posterior probability.
Now we are ready to establish the contraction rate. The proof of the bounds needed to establish this one is similar to that of Proposition 5.1 of Qiao and Polonik (2016). Let DΥx0 ,V f (t) :=
∇f (Υx0 (t))T V (Υx0 (t)) and
2
DΥ
x

0 ,V

f (t) := h∇h∇f (Υx0 (t)), V (Υx0 (t))i, V (Υx0 (t))i
= ∇f (Υx0 (t))T ∇V (Υx0 (t))V (Υx0 (t)) + V (Υx0 (t))T Hf (Υx0 (t))V (Υx0 (t)).

We have for some t̃ between tx0 and t∗x0 ,
2
0 = DΥx0 ,V f (tx0 ) = DΥx0 ,V f (t∗x0 ) + DΥ
x

0 ,V

f (t̃)(t̃ − t∗x0 ).

We claim that
2
Π( inf |DΥ
x
x0 ∈G

P

0 ,V

0
f (t̃)| > η|Y ) −→
0.

27

Since
2
sup DΥ
x
x∈G

0 ,V

2
f (t̃) − DΥ
∗
x

0

,V ∗ f

∗

2
and by Assumption (A3), inf x0 ∈G |DΥ
∗
x

(t∗x0 ) ≥

0

,V ∗ f

2
inf |DΥ
x

x0 ∈G

∗ (t∗ )|
x0

0 ,V

2
f (t̃)| − inf |DΥ
∗
x
x0 ∈G

0

,V ∗ f

∗

(t∗x0 )|

≥ η, it suffices to show that with P0 -probability

2
2
∗ ∗
tending to 1, supx∈G DΥ
f (t̃) − DΥ
∗ ,V ∗ f (tx0 ) is small with high posterior probability. To see
x0 ,V
x0
this, we write
2
sup DΥ
x
x

0 ,V

2
f (t̃) − DΥ
∗
x

0

,V ∗ f

∗

2
(t∗x0 ) ≤ sup DΥ
x
x

2
+ sup DΥ
∗
x

0 ,V

0

x

2
f (t̃) − DΥ
∗
x

,V ∗ f

0

∗

,V ∗ f

2
(t̃) − DΥ
∗
x

0

∗

(t̃)

,V ∗ f

∗

(t∗x0 ) .

The first term contracts to zero, simply due to the uniform contraction results for ∇f (x), V (x), ∇V (x),
Hf (x) (Lemma 8.2) and for Υx0 (Theorem 5.1). The second term contracts to zero, since Υ∗x0 (t) is
continuous in t and ∇f ∗ (x), V ∗ (x), ∇V ∗ (x), Hf ∗ (x) are uniform continuous.
Also, supx0 |DΥx0 ,V f (t∗x0 )| = supx0 |DΥx0 ,V f (t∗x0 ) − DΥ∗x ,V ∗ f (t∗x0 )| is given by
0

sup h∇f (Υx0 (t∗x0 )) − ∇f ∗ (Υ∗x0 (t∗x0 )), V (Υx0 (t∗x0 )) − V ∗ (Υ∗x0 (t∗x0 ))i
x0

+ sup h∇f (Υx0 (t∗x0 )) − ∇f ∗ (Υ∗x0 (t∗x0 )), V ∗ (Υ∗x0 (t∗x0 ))i
x0

+ sup h∇f ∗ (Υ∗x0 (t∗x0 )), V (Υx0 (t∗x0 )) − V ∗ (Υ∗x0 (t∗x0 ))i .
x0

Now since
sup |tx0 − t∗x0 | ≤
x0

1
2
inf x0 DΥ
x

0

,V f (t̃)

sup |DΥx0 ,V f (t∗x0 )|,
x0

we have

sup |t∗x0 − tx0 | . sup k∇f (Υx0 (t∗x0 )) − ∇f ∗ (Υ∗x0 (t∗x0 ))k sup kV ∗ (Υ∗x0 (t∗x0 ))k

x0 ∈G

x0 ∈G

+ sup kV
x0 ∈G

x0 ∈G

(Υx0 (t∗x0 ))

−V

∗

(Υ∗x0 (t∗x0 ))k



sup k∇f ∗ (Υ∗x0 (t∗x0 ))k.

x0 ∈G

It is easy to see that
sup k∇f (Υx0 (t∗x0 )) − ∇f ∗ (Υ∗x0 (t∗x0 ))k . sup k∇f (x) − ∇f ∗ (x)k + sup

x0 ∈G

sup kV (Υx0 (t∗x0 )) − V ∗ (Υ∗x0 (t∗x0 ))k . sup kV (x) − V ∗ (x)k + sup

x0 ∈G

sup kΥx0 (t) − Υ∗x0 (t)k,

x0 ∈G t∈[0,Tx∗ ]
0

x0 ∈G

sup kΥx0 (t) − Υ∗x0 (t)k.

x0 ∈G t∈[0,Tx∗ ]
0

x0 ∈G

With the choice of J  (n/ log n)1/(1+2α) , by Lemma 8.2, supx k∇f (x)−∇f ∗ (x)k has posterior contraction rate of order (n/ log n)(3−2α)/(2(1+2α)) , while supx kV (x)−V ∗ (x)k has that of order (n/ log n)(5−2α)/(2(1+2α)) .
Therefore, considering the rate from Theorem 5.1, we have the desired result.

28

Proof of Theorem 5.3 . One can write
Υx0 (tx0 ) − Υ∗x0 (t∗x0 ) = Υx0 (tx0 ) − Υx0 (t∗x0 ) + Υx0 (t∗x0 ) − Υ∗x0 (t∗x0 )
= V (Υx0 (t̃x0 ))(tx0 − t∗x0 ) + Υx0 (t∗x0 ) − Υ∗x0 (t∗x0 )
= V ∗ (Υ∗x0 (t∗x0 ))(tx0 − t∗x0 ) + (V (Υx0 (t̃x0 )) − V ∗ (Υ∗x0 (t∗x0 )))(tx0 − t∗x0 )
+ Υx0 (t∗x0 ) − Υ∗x0 (t∗x0 )
for t̃x0 between tx0 and t∗x0 . Therefore,


sup kΥx0 (tx0 ) − Υ∗x0 (t∗x0 )k . sup |tx0 − t∗x0 | 1 + sup kV (Υx0 (t̃x0 )) − V ∗ (Υ∗x0 (t∗x0 ))k
x0 ∈G

x0 ∈G

x0 ∈G

+ sup kΥx0 (t∗x0 ) − Υ∗x0 (t∗x0 )k.
x0 ∈G

In view of the posterior contraction of V (Lemma 8.2), Theorem 5.1 and Proposition 5.2, the posterior
of supx0 ∈G kV (Υx0 (t̃x0 )) − V ∗ (Υ∗x0 (t∗x0 ))k contracts to zero.
Since
sup kΥx0 (tx0 ) − Υ∗x0 (t∗x0 )k . sup |tx0 − t∗x0 | + sup kΥx0 (t∗x0 ) − Υ∗x0 (t∗x0 )k,

x0 ∈G

x0 ∈G

x0 ∈G

by Theorem 5.1 and Proposition 5.2, we establish the posterior contraction rate for supx0 ∈G kΥx0 (tx0 )−
Υ∗x0 (t∗x0 )k.
Proof of Proposition 5.6. Consider (A2) for (i). Note that
λ(Υx0 (tx0 )) − λ∗ (Υ∗x0 (t∗x0 ))
=V (Υx0 (tx0 ))T Hf (Υx0 (tx0 ))V (Υx0 (tx0 )) − V ∗ (Υ∗x0 (t∗x0 ))T Hf ∗ (Υ∗x0 (t∗x0 ))V ∗ (Υ∗x0 (t∗x0 )).
Thus by the posterior uniform contraction and uniform continuity of V , Hf (Lemma 8.2) and the uniform contraction of Υx0 (tx0 ) (Theorem 5.3) over x0 , with P0 -probability tending to 1, clearly condition
(A2) holds with high posterior. Condition (A3) trivially holds by the same uniform contraction results
and noting that
h∇h∇f (Υx0 (tx0 )), V (Υx0 (tx0 ))i, V (Υx0 (tx0 ))i
=∇f (Υx0 (tx0 ))T ∇V (Υx0 (tx0 ))V (Υx0 (tx0 )) + V (Υx0 (tx0 ))T Hf (Υx0 (tx0 ))V (Υx0 (tx0 )).
The compactness follows trivially by the continuity of λ(x) and h∇f (x), V (x)i and the fact that intersection of closed sets is closed and the boundedness of [0, 1]2 . Condition (A5) holds also due to
Theorem 5.3. In view of Lemma 8.4 and Theorem 5.4, (ii) holds.
Proof of Lemma 5.7 . Let x0 be an arbitrary point on L̂. Thus Υx0 (0) = x0 ∈ L̂ and Υx0 (tx0 ) ∈ L
for some tx0 > 0. Note that inf y∈L kx0 − yk ≤ kΥx0 (0) − Υx0 (tx0 )k ≤ tx0 as kV (x)k = 1. Let
d
DΥx0 ,V f (t) := dt
f (Υx0 (t)) = ∇f (Υx0 (t))T V (Υx0 (t)) and
2
DΥ
x

0 ,V

f (t) := h∇h∇f (Υx0 (t)), V (Υx0 (t))i, V (Υx0 (t))i
= ∇f (Υx0 (t))T ∇V (Υx0 (t))V (Υx0 (t)) + V (Υx0 (t))T Hf (Υx0 (t))V (Υx0 (t)),
29

where the second equality is due to the chain rule. A Taylor expansion yields that
 2
f (t̃)
DΥx0 ,V f (t) − DΥx0 ,V f (tx0 ) = t − tx0 DΥ
x ,V
0

for some t̃ between 0 and tx0 . In particular, since DΥx0 ,V f (tx0 ) = 0, letting t = 0, we obtain
2
DΥx0 ,V f (0) = −tx0 DΥ
x

0 ,V

f (t̃).

Furthermore,
|DΥx0 ,V f (0)| = |DΥx0 ,V f (0) − DΥx

0 ,V̂

fˆ(0)|

= |∇f (Υx0 (0))T V (Υx0 (0)) − ∇fˆ(Υx0 (0))T V̂ (Υx0 (0))|
≤ sup |∇f (x)T V (x) − ∇fˆ(x)T V̂ (x)|
x

≤ sup |∇f (x)T (V (x) − V̂ (x))| + sup |(∇f (x) − ∇fˆ(x))T V̂ (x))|
x

x

≤ C(sup kV (x) − V̂ (x)k + sup k∇f (x) − ∇fˆ(x)k)
x

x

≤ C sup kV (x) − V̂ (x)k.
x

By the uniform continuity of ∇f (x), ∇V (x), V (x) and Hf (x) and the continuity of Υx0 (t) in t, without
loss of generality, we can make tx0 small enough (see comments after the proof). Thus we have
2
DΥ
x

0 ,V

2
By Assumption (A3), DΥ
x
2
DΥ
x

0 ,V

0 ,V

2
f (t̃) − DΥ
x

0 ,V

f (tx0 ) < η/2.

f (tx0 ) > η, and hence

2
f (t̃) ≥ DΥ
x

0 ,V

2
f (tx0 ) − DΥ
x

0 ,V

2
f (t̃) − DΥ
x

0 ,V

f (tx0 ) > η − η/2 = η/2.

Therefore, inf y∈L kx0 − yk ≤ tx0 ≤ Cη supx kV (x) − V̂ (x)k. Thus d(L̂|L) ≤ Cη supx kV (x) − V̂ (x)k.
Similarly, d(L|L̂) ≤ Cη supx kV (x) − V̂ (x)k. Therefore, Haus(L, L̂) ≤ Cη supx∈[0,1]2 kV (x) − V̂ (x)k. Recall that V (x) = G(df 2 (x)). Now since G is a fixed Lipschitz continuous function, it is easy to get
the upper bound for supx∈[0,1]2 kV (x) − V̂ (x)k in terms of the supremum distance of the derivatives of
f (x) − fˆ(x).
In above proof, tx0 can be made arbitrarily small in the limit. To see this, if Assumption (A5) holds
for the f (or more precisely, Υx0 ), then kΥx0 (tx0 )− Υ̂x0 (t̂x0 )k = kΥx0 (tx0 )−x0 k = kΥx0 (tx0 )−Υx0 (0)k >
CG tx0 . Since kΥ̂x0 (t̂x0 ) − Υx0 (tx0 )k can be made arbitrarily small due to the closedness in supremum
norm (see previous theorems), tx0 can be made arbitrarily small.
Proof of Theorem 5.9. For γ < 1/2, by argument in the proof of Theorem of 5.3 of Yoo and Ghosal
(2016), one can establish that


Rn,k,γ  E sup kf (k) − f˜(k) k ,
x

2
Rn,k,γ

 (log n/n)J 6  (log n/n)(α−2)/(α+1) .
30

Recall from (2), Π(f (k) |Y, σ 2 )|σ2 =σ̂2 ∼ GP(f˜(k) , σ̂ 2 Σ(k) ). By Borell’s inequality (see Proposition
A.2.1 of Van der Vaart and Welnner (1996)),
Π(L ∈
/ CL |Y, σ̂ 2 ) ≤ Π(L corresponds to f ∈
/ Cf,k,γ for some k ∈ {1, 2, 3}|Y, σ̂ 2 )
≤

3
X

Π(sup |f (k) − f˜(k) | > ρRn,k,γ |Y, σ̂ 2 )

k=1

x

2
/cn,k ],
≤ 3 max exp[−(ρ − 1)2 Rn,k,γ
1≤k≤3

where cn,k = supx var(f (k) − f˜(k) |Y, σ̂ 2 ) is bounded by a constant multiple of
(k)

−1
−3/(α+1) (2−α)/(α+1)
n
.
sup Σ(k) (x, x) . sup kbJ1 ,J2 (x)k2 k(B T B + Λ−1
0 ) k(2,2) . (log n)
x

x

Therefore, the above posterior probability tends to zero. Finally, P0 (L∗ ∈ CL ) = P0 (supx |f ∗(k) −f˜(k) | ≤
ρRn,k,γ , ∀k ∈ {1, 2, 3}) → 1, by Lemma 8.4, establishing the coverage of CL .
To see CL ⊂ C̄L , for any L ∈ CL , it is induced by some f such that supx kf (k) − f˜(k) k ≤
ρ max1≤k≤3 Rn,k,γ , ∀k ∈ {1, 2, 3}. In view of the discussion in the beginning of this section and Proposition 5.6, since Lemma 5.7 holds with P0 -probability tending to 1 with L being the filament in posterior
and L̂ being the filament induced by the posterior mean f˜, the result immediately follows.

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