text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
E +
)δξ = ((Ex, Ey ) · (Rp, Rq))δξ =⇒ δE = 1.
∂R
∂x ∂p
∂E ∂R
∂y ∂q
1.5 Generating an Initial Curve
While only having to measure a single initial curve for profiling is far better than having to measure an entire surface, it is still
undesirable and is not a caveat we are satisfied with for this solution. Below, we ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/617445f0e31836831b40d42cb2f11a10_MIT6_801F20_lec10.pdf |
any special points on the object where we already
know the orientation without a measurement? These points are along the edge, or occluding boundary, of our objects of
interest. Here, we know the surface normal of each of these points. Could we use these edge points as “starting points” for our
SfS solution?
It tur... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/617445f0e31836831b40d42cb2f11a10_MIT6_801F20_lec10.pdf |
Additionally, if we have stationary brightness points in image space, we encounter the “dual” of this problem. By definition
stationary points in the image space imply that:
∂E
∂x
=
∂E
∂y
= 0
6
This in turn implies that p and q cannot be stepped:
dp
dξ
dq
dξ
=
=
∂E
∂x
∂E
∂y
=... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/617445f0e31836831b40d42cb2f11a10_MIT6_801F20_lec10.pdf |
circle in the plane centered at the stationary point with
radius - this means all points in this plane will have the same surface orientation as the stationary point. Note that
mathematically, a local 2D plane on a 3D surface is equivalent to a 2-manifold [1]. This is good in the sense that we know
the surface ori... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/617445f0e31836831b40d42cb2f11a10_MIT6_801F20_lec10.pdf |
shape? It turns out the answer is no, again because of stationary points.
But if we look at the second derivatives of brightness:
=
(8x) = 8
=
(32y) = 32
Exx =
Eyy =
Exy =
∂2E
∂x2
∂2E
∂y2
∂2E
∂x∂y
∂E
∂x
∂E
∂y
∂
∂x
=
(32y) =
∂
∂y
(8y) = 0
7
These second derivatives, as we will discuss more in t... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/617445f0e31836831b40d42cb2f11a10_MIT6_801F20_lec10.pdf |
Chapter 5
Coupled Fluids with Heat and Mass
Transfer
5.1 November 26, 2003: Coupled Fluids, Heat and Mass Transfer!
Mechanics:
• Congrats to Jenny and David for winning the contest, prize: $5 Tosci’s.
• PS8 on Stellar, due Fri 12/5.
• Evaluations next Wednesday 12/3.
Muddy from last time:
• Time smoothing: what... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
T
x
δT = 3.6
�
αx
U∞
3.6
= �
U∞ x
α
=
√
3.6
RexPr
When is flow uniform? In a solid, or for much larger thermal boundary layer than fluid, so α >> ν, Pr<< 1.
66
Another way to look at it:
Large Prandtl number (>.5) means
δT = 0.72Pr−1/2
δu
δT
= 0.975Pr−1/3
δu
Liquid metals (and about nothing else) hav... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
∞)
�
�
U
1
∞ = hx(Ts − T∞)
√
π αx
�
k
hx = √
π αx
U∞ =
∞
kρcpU
πx
Likewise for mass transfer, ρcp is effectively one, so:
hDx =
�
DU
∞
πx
Next time: average, dimensional analysis, δT < δu case.
67
5.2 December 1, 2003: Nusselt Number, Heat and Mass Transfer
Coefficients
Mechanics:
• Evals Wednesday.
... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
is ratio of square roots of diffusivity, which is
inverse sqrt(Pr).
• Case 2: smaller thermal(/concentration) boundary layer (Pr>5 or so): consider T/C BL to have linear
velocity, smaller velocity means thicker T/C BL. Here:
• Moving on, back to case 1, calculated q|y=0 from erf solution:
δC /δu or δT /δu = 0.975Pr... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
=
L/Dsolid =
1/h
Resistance to conduction in solid
Resistance due to BL in liquid
Uses L=solid thickness, Dsolid. Heat transfer note: you get one extra dimensionless number, due to heating
by viscous friction.
Here, Nusselt #, L=length of plate (in flow direction), the conduction and BL are in the same medium,
u... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
heat transfer
coefficient, kinetic energy transfer coefficient. Types: local, global/average. Laminar flow variation:
both∼ 1/
x. Laminar fL = 2fx x=L, hL = 2hx x=L. Dimensionless:
f = f (Re), Nu=f (Re,Pr). Different correlations for different geometries.
x, average∼ 1/
x, integral∼
√
√
√
|
|
• Other Nusselt numbers from ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
dL
L dT
β = V
= 3α.
=
=
ρ =
P
RT
, β = −
1 dρ
ρ dT
= −
RT
P
−
P
RT 2
= 1/T.
�
�
Also βC =
−
1 dρ
ρ dC .
Simplest case: vertical wall, Ts at wall, T∞ with density ρ∞ away from it, x vertical and y horizontal for
consistency with forced convection BL. Assume:
1. Uniform kinematic viscosity ν = ν∞.
2. S... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
, this gives driving force in the positivex direction, which is up, like it’s supposed
to. Okay, that’s all for today, more next time.
71
5.4 December 5: Wrapup Natural Convection
Mechanics:
• Test 2: before max=90, mean 75.38, std. dev 12.23; after max=100, mean 95.76, std. dev 6.37.
Muddy from last time:
• D... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
∞)/(Ts − T∞), dimensionless ux = Rex/2
Grx on P&G
p. 232 corresponding to dimensional graphs in W3R p. 313. Explain velocity BL is always at least as thick
as thermal BL, but thermal can be thinner for large Pr.
√
√
√
Grx vs. y/ 4
x
Forced convection: δ ∝
√
Natural convection: δ ∝ 4
x
Note: in P&G p. 232 plots, Pr... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
494Pr2/3)2/5
Again, velocity0.8 in a way, sorta like turbulent forced convection boundary layers.
73
5.5 December 8: Wrapup Natural Convection, Streamfunction
and Vorticity
Mechanics:
• Final exam Monday 12/15 in 4149. Discuss operation, incl. closed/open sections, new diff eq, essay.
Muddy from last time:
• W... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
�ν
2
Grx
=
1 uxx
2 ν
�
ν2
gβΔT x3
=
√
ux
gβΔT x
⇒
ux,max = f (Pr) gβΔT x.
�
�
4
Grx
4
δu
x
= f (Pr) ⇒ δu = f (Pr) �
4
x
Grx/4
√
=
�
2f (Pr) 4
x4ν2
gβΔT x3
√
=
�
xν2
gβΔT
.
2f (Pr) 4
These two results are consistent with: ux,max ∝ thickness2, forced convection Δux/Δy goes as 1/
Rex.
√
Other g... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
, combining, annihilating.
Other application: crystal rotation in semisolid rheology.
Stream function, for incompressible flow where � · �u = 0:
ux =
∂Ψ
∂y
, uy = −
∂Ψ
∂x
2
Collapses velocity components into one parameter. Look at Ψ = Ax, Ψ = By, Ψ = Ax + By, Ψ2 = x2 + y .
Cool.
74
Gradient is normal to flow ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
okes; I like to think mine is more straightforward,
but you can read W3R if needed.
Also called “inviscid flow”. Motivation: tub with hole, pretty close to zero friction factor, velocity is
infinity? No. Something other than viscosity limits it.
NavierStokes, throw out viscous terms:
ρ
D�u
Dt
= −� p + ρ�g
Change... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
in corner, 1 ρV 2 + P1
at base over spout, 2 ρV 2 − ρgh2 at tube end. Three equations in three unknowns. Solves to P1 = ρgh,
V 2 = 2g(h + h2), P1 = − ρgh2. Can also fill in the table...
2
1
Conditions:
• No shear or other losses (not nearly fullydeveloped)
• No interaction with internal solids, etc.
• No heat in ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
Shameless plug...)
Thank Albert for a terrific job as a TA!
Last muddy questions
• What is the relevance of the boundary layer thickness to the Bernoulli equation? The boundary layer
is a region where there is quite a bit of shear, and sometimes turbulence. If it is thin relative to the size
of the problem (e.g. re... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
CA
CA,out
CA,in
= exp
−
�
�
hD A
t
V
Two extremes in continuous reactor behavior with flow rate Q: plug flow and perfect mixing.
77
Plug flow is like a minibatch with tR = V /Q, draw plug in a pipe, derive:
With a surface, the V s cancel, left with
CA,out
CA,in
CA,out
CA,in
�
�
−
kV
Q
= exp
�
= e... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
elmaking: batch, but folk want to make continuous.
78 | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6185b0ff143062ed97aea16a782c8603_chap5.pdf |
18.S997: High Dimensional Statistics
Lecture Notes
(This version: July 14, 2015)
Philippe Rigollet
Spring 2015
Preface
These lecture notes were written for the course 18.S997: High Dimensional
Statistics at MIT. They build on a set of notes that was prepared at Princeton
University in 2013-14.
Over the past decade, st... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
that Donoho and Johnstone have made
the first contributions on this topic in the early nineties.
i
Preface
ii
Acknowledgements. These notes were improved thanks to the careful read-
ing and comments of Mark Cerenzia, Youssef El Moujahid, Georgina Hall,
Jan-Christian Hu¨tter, Gautam Kamath, Kevin Lin, Ali Makhdoumi, Yar... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
subGd(σ2)
subE(σ2)
Ber(p)
Bin(n, p)
Lap(λ)
PX
∈
IR and variance σ2 > 0
IRd
d
×
Univariate Gaussian distribution with mean µ
IRd and covariance matrix Σ
d-variate distribution with mean µ
Univariate sub-Gaussian distributions with variance proxy σ2 > 0
d-variate sub-Gaussian distributions with variance proxy σ2 > 0
sub-... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
. . . . . . .
2.5 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Misspecified Linear Models
. . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Oracle inequalities
3.2 Nonparametric regression . . . . . . . . . . . . . . . . . . . . .
3.3 Problem Set . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
. . . . . . . 114
Bibliography
121
Introduction
This course is mainly about learning a regression function from a collection
of observations.
In this chapter, after defining this task formally, we give
an overview of the course and the questions around regression. We adopt
the statistical learning point of view wh... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
Formally, the regression function of Y onto X is defined by:
f (x) = IE[Y
|
X = x] ,
x
.
∈ X
As we will see, it arises naturally in the context of prediction.
Best prediction and prediction risk
Suppose for a moment that you know the conditional distribution of Y given
X. Given the realization of X = x, your goa... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
of g can be decom
X →
IE[Y
g(X)]2 = IE[Y
f (X) + f (X)
−
= IE[Y
−
f (X)]2 + IE[f (X)
g(X)]2
g(X)]2 + 2IE[Y
−
−
The cross-product term satisfies
−
f (X)][f (X)
g(X)]
−
−
IE[Y
−
f (X)][f (X)
−
g(X)] = IE
IE
[Y
= IE
= IE
[IE(Y
[
(
|
[f (X)
[
−
X)
−
f (X)][f (X)
g(X)]
X
f (X)][f (X)
g(X)]
−
f (X)][f (X)
−
g(X)]
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
we observe a sample
that consists
of independent copies of (X, Y ). The goal of regression function estimation is
to use this data to construct an estimator fˆ n :
that has small L2 risk
R(fˆ n).
(X1, Y1), . . . , (Xn, Yn)
}
Dn =
X → Y
{
Let PX denote the marginal distribution of X and for any h :
define
Note ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
equivalent to study the
2 and
estimation error
I2
R(fˆ n) are random quantities and we need deterministic summaries to quantify
their size. It is customary to use one of the two following options. Let
φn}n
{
be a sequence of positive numbers that tends to zero as n goes to infinity.
2. Note that if fˆ n is random... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
arbitrary and can be replaced by another positive constant.
fˆ n
2
Such bounds control the tail of the distribution of
f
2. They show
−
I
fˆ nI
2
2 can be. Such
how large the quantiles of the random variable
bounds are favored in learning theory, and are sometimes called PAC
bounds (for Probably Approximately Co... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
X)]2 +
fˆ n
I
−
−
2 .2
f
I
fˆ n
f
I
−
2
2 as a measure of
This equality allowed us to consider only the part
error. While this decomposition may not hold for other risk measures, it may
be desirable to explore other distances (or pseudo-distances). This leads to two
distinct ways to measure error. Either by bou... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
It generalizes both the L2 distance and the sup-norm error by
taking for any p
1, the pseudo distance
≥
dp(fˆ n, f ) =
fˆ n
|
−
|
f
pdPX .
1
X
The choice of p is somewhat arbitrary and mostly employed as a mathe
matical exercise.
Note that these three examples can be split into two families: global (Sup-norm ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
n and θ as long as θ is identifiable.
X
IRd .
∈
ˆ
MODELS AND METHODS
Empirical risk minimization
In our considerations on measuring the performance of an estimator fˆ n, we
have carefully avoided the question of how to construct fˆ n. This is of course
one of the most important task of statistics. As we will see... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
to from the empirical
risk of g defined by
−
Rn(g) =
n1
n
g(Xi)
2
.
Yi
−
i=1
n (
We can now proceed to minimizing this risk. However, we have to be careful.
0 for all g. Therefore any function g such that Yi = g(Xi) for
Indeed, Rn(g)
all i = 1, . . . , n is a minimizer of the empirical risk. Yet, it may not b... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
0.8
x
x
x
Figure 1. It may not be the best choice idea to have fˆn(Xi) = Yi for all i = 1, . . . , n.
small). In both cases, this extra knowledge can be incorporated to ERM using
either a constraint :
or a penalty:
or both
min
g
min
g
∈G
min Rn(g)
g
∈G
Rn(g) + pen(g)
,
{
}
Rn
(g) + pen(g)
,
{
}
These schem... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
Introduction
Linear models
7
X
= IRd, an all time favorite constraint
is the class of linear functions
When
IRd . Under
that are of the form g(x) = x⊤θ, that is parametrized by θ
this constraint, the estimator obtained by ERM is usually called least squares
estimator and is defined by fˆ n(x) = x⊤θˆ, where
∈ ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
about misspecified model, i.e., we try to fit a linear model to data
that may not come from a linear model. Since linear models can have good
approximation properties especially when the dimension d is large, our hope is
that the linear model is never too far from the truth.
,
·
In the case of a misspecified model, t... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
−
−
G
I
I
I
I
I
f
IE
I
fˆ n
f
2
I2
−
≤ I
¯
f
f
2 + φn .
I2
−
¯
The above inequality is called an oracle inequality. Indeed, it says that if φn
is small enough, then fˆ n the estimator mimics the oracle f . It is called “oracle”
because it cannot be constructed without the knowledge of the unknown f . It
is clear... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
estimators. With the development of aggregation
[Nem00, Tsy03, Rig06] and high dimensional statistics [CT07, BRT09, RT11],
they have become important finite sample results that characterize the inter
play between the important parameters of the problem.
In some favorable instances, that is when the Xis enjoy specific... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
informal discussion here.
∈
As we will see in Chapter 2, if the regression function is linear f (x) = x
⊤θ∗
,
IRd, and under some assumptions on the marginal distribution of X, then
θ∗
the least squares estimator fˆ n(x) = x⊤θˆn satisfies
∈
fˆ n
IE
I
f
2
I2
−
≤
C
d
n
,
(1)
where C > 0 is a constant and in ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
1
n
1I(θj = 0) .
=
6
Introduction
9
Sparsity is just one of many ways to limit the size of the set of potential
θ vectors to consider. One could consider vectors θ that have the following
structure for example (see Figure 2):
•
•
•
•
Monotonic: θ1 ≥
θj
Smooth:
θi
| ≤
|
Piecewise constant:
−
θd
θ2 ≥ ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
1.
coefficients
mathematical ways to capture this phenomena, including ℓq -“balls” for q
For q > 0, the unit ℓq-ball of IRd is defined as
θj
|
≤
θ
|
|
Bq (R) =
θ
∈
IRd :
θ
q =
q
|
|
{
d
j=1
n
q
θj
|
|
≤
1
}
3
|
θ
where
vectors in the unit ℓq ball can be approximated by sparse vectors.
k
Note that the set... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
))
Therefore, the price to pay for not knowing which subspace to look at is only
a logarithmic factor.
n
(
≃
(
)
(
3Strictly speaking, |θ|q is a norm and the ℓq ball is a ball only for q ≥ 1.
6
Introduction
10
θj
θj
Monotone
Smooth
θj
j
j
Piecewise constant
Smooth in a different basis
θj
j
j
Figure 2. Exampl... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
is also unknown as it depends on the unknown PX . This is absolutely
αk}k
∈
Z
{
∈
infinite sequence
ϕk}k
{
∈
Introduction
11
correct but we will make the convenient assumption that PX is (essentially)
known whenever this is needed.
|
|
k
Even if infinity is countable, we still have to estimate an infinite number ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
Indeed, for any cut-off
> k0, but rather that the sequence
−
|
C
k
k
{
{
{
|
|
|
|
k0, define the oracle
¯fk0 =
αkϕk .
k0
k
n
|≤
|
Note that it depends on the unknown αk and define the estimator
fˆ n =
αˆkϕk ,
k0
k
n
|≤
where ˆαk are some data-driven coefficients (obtained by least-squares for ex
ample). Then b... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
see that we can strike a com
promise called bias-variance tradeoff.
(ˆαk
−
|≤
k0
k
The main difference here with oracle inequalities is that we make assump
tions on the regression function (here in terms of smoothness) in order to
4Here we illustrate a convenient notational convention that we will be using through
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
see that even if the smoothness index γ is unknown, we can select k0
in a data-driven way that achieves almost the same performance as if γ were
known. This phenomenon is called adaptation (to γ).
It is important to notice the main difference between the approach taken
in nonparametric regression and the one in spar... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
signal
and noise, and that satisfy
M = S + N .
Here N is a random matrix such that IE[N ] = 0, the all-zero matrix. The goal
is to estimate the signal matrix S from the observation of M .
The structure of S can also be chosen in various ways. We will consider the
case where S is sparse in the sense that it has ma... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
task is much easier and is dominated by the former in terms of statistical price.
Another important example of matrix estimation is high-dimensional co
variance estimation, where the goal is to estimate the covariance matrix of a
IRd, or its leading eigenvectors, based on n observations.
random vector X
Such a pro... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
precisely, we can prove that for any estimator
f˜ n, there exists a function f of the form f (x) = x⊤θ∗ such that
fˆ n
IE
I
f
I
−
2
2 > c
d
n
for some positive constant c. Here we used a different notation for the constant
to emphasize the fact that lower bounds guarantee optimality only up to a
constant factor.... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
1.1 GAUSSIAN TAILS AND MGF
Recall that a random variable X
IR has Gaussian distribution iff it has a
density p with respect to the Lebesgue measure on IR given by
∈
p(x) =
1
2πσ2
√
exp
(x
µ)2
−
2σ2
−
,
x
IR ,
∈
(
∈
∼ N
)
IR and σ2 = var(X) > 0 are the mean and variance of
where µ = IE(X)
(µ, σ2). Note th... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
� √
t2
2σ2
−
t
.
14
1.1. Gaussian tails and MGF
15
Figure 1.1. Probabilities of falling within 1, 2, and 3 standard deviations close to the
mean in a Gaussian distribution. Source http://www.openintro.org/
and
X
IP(
|
−
µ
|
> t)
≤
2 e −
π
t2
2σ2
t
.
Proof. Note that it is sufficient to prove the theorem ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
M : s
→
M (s) = IE[exp(sZ)] .
1.2. Sub-Gaussian random variables and Chernoff bounds
16
Indeed in the case of a standard Gaussian random variable, we have
M (s) = IE[exp(sZ)] =
1
√
2π
1
√
2π
2 s
2= e
=
.
sz
e e
−
2
z
2 dz
(z−s)2
2
+ s 2
2 dz
e
−
1
1
It follows that if X
∼ N
(µ, σ2), then I... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
2 log(2/δ)
n
,
(1.1)
This is almost the confidence interval that you used in introductory statistics.
The only difference is that we used an approximation for the Gaussian tail
whereas statistical tables or software use a much more accurate computation.
Figure 1.2 shows the ration of the width of the confidence inte... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
∈
1.2. Sub-Gaussian random variables and Chernoff bounds
17
Figure 1.2. Width of confidence intervals from exact computation in R (red dashed)
and (1.1) (solid black).
proxy σ2 if IE[X] = 0 and u⊤X is sub-Gaussian with variance proxy σ2 for
d
subGd(σ2). A ran
any unit vector u
−
T is said to be sub-Gaussia... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
generating function into a tail bound. Using Markov’s inequality, we have for
any s > 0,
∼
(
)
IP(X > t)
≤
IP
e sX > est
IE
sX
e
est
≤
.
Next we use the fact that X is sub-Gaussian to get
(
)
IP(X > t)
≤
σ2 s 2
2
e
st
.
−
1.2. Sub-Gaussian random variables and Chernoff bounds
18
The above inequality hol... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
σ2) can be bounded by those of
show that the absolute moments of X
Z
(0, σ2) up to multiplicative constants.
∼
∼ N
Lemma 1.4. Let X be a random variable such that
IP[
X
|
|
> t]
≤
2 exp
t2
2σ2
,
)
−
(
then for any positive integer k
1,
≤
≥
k]
|
IE[
X
|
(2σ2)k/2kΓ(k/2) .
In particular,
and IE[
X
]
|
|
≤
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
/eσ
e
√
k .
(
Moreover, for k = 1, we have
)
2Γ(1/2) = 2π.
√
r
√
Using moments, we can prove the following reciprocal to Lemma 1.3.
Lemma 1.5. If (1.3) holds, then for any s > 0, it holds
IE[exp(sX)]
4σ2 s
2
e
.
≤
As a result, we will sometimes write X
subG(σ2) when it satisfies (1.3).
∼
Proof. We use the T... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
/2(2k + 1)Γ(k + 1/2)
(2k + 1)!
2(k!)2
(2k)!
≤
From the above Lemma, we see that sub-Gaussian random variables can
be equivalently defined from their tail bounds and their moment generating
functions, up to constants.
Sums of independent sub-Gaussian random variables
Recall that if X1, . . . , Xn are i.i.d
(0, ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
ollary 1.7. Let X1, . . . , Xn be n independent random variables such that
Xi
subG(σ2). Then for any a
IRn, we have
∼
∈
n
IP
i=1
[ �
n
and
aiXi > t
i
≤
exp
−
(
t2
2σ2
a
|
,
2
2
|
)
IP
aiXi <
i=1
[ �
exp
−
t
i
≤
t2
− 2σ2
a
|
2
2
|
)
(
Of special interest is the case where ai = 1/n for all i. Then, we ge... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
(b−a)2
8
e
.
In particular, X
subG(
(b
a)2
−
4
) .
∼
1.2. Sub-Gaussian random variables and Chernoff bounds
21
Proof. Define ψ(s) = log IE[esX ], and observe that and we can readily compute
ψ ′ (s) =
IE[XesX ]
IE[esX ]
,
ψ ′′ (s)
=
IE[X 2e
sX ]
IE[esX ]
IE[XesX ]
IE[esX ]
2
.
�
−
�
Thus ψ ′′ (s) ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
such that almost surely,
Let X¯ =
1
n
n
i=1
Xi, then for any t > 0,
Xi
∈
[ai, bi] ,
i.
∀
�
P
¯
IP( X
−
IE( X) > t)
¯
exp
≤
−
(
2t2
2n
n (bi
i=1
ai)2
−
,
)
and
¯
IP( X
−
¯
IE( X) <
t)
−
≤
exp
�
P
−
2 2
tn
2
n (bi
i=1
−
ai)2
.
)
(
Note that Hoeffding’s lemma is for any bounded random variables. For
�
P
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
EXPONENTIAL RANDOM VARIABLES
What can we say when a centered random variable is not sub-Gaussian?
A typical example is the double exponential (or Laplace) distribution with
parameter 1, denoted by Lap(1). Let X
Lap(1) and observe that
∼
X
IP(
|
|
> t) = e
−
t
,
0 .
t
≥
In particular, the tails of this distribut... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
(cid:3)
(cid:2)
Lemma 1.10. Let X be a centered random variable such that IP(
|
2e−
2t/λ for some λ > 0. Then, for any positive integer k
1,
≥
> t)
X
|
≤
IE[
X
|
k]
|
≤
λk k! .
Moreover,
IE[
X
|
k])1/k
|
≤
2λk ,
and the moment generating function of X satisfies
(
IE e sX
e 2s 2λ2
,
≤
s
∀|
| ≤
1
2λ
.
Pr... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
2λ
s
(
|
| ≤
sX
IE e
(cid:2)
(cid:3)
1 +
1 +
≤
≤
= 1 + s
∞
k=2
�
X
∞
(
|
k=2
�
X
2λ2
s
|
kIE[
|
|
k!
X
k]
|
λ)k
s
|
λ)k
∞
(
|
k=0
X
�
s
|
1 + 2s 2λ2
2λ2
2s
e
≤
≤
This leads to the following definition
1
| ≤ 2λ
s
|
Definition 1.11. A random variable X is said to be sub-exponential with
subE(λ)) i... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
sk2k
−
1
(cid:3)
IE[X 2k] + (IE[X 2])k
(
k=2
�
X
∞ sk4kIE[X 2k]
k!
)
1 +
1 +
1 +
≤
≤
≤
k=2
X
�
∞
2(k!)
sk4k2(2σ2)kk!
2(k!)
∞
(8sσ2)k
k=0
�
X
k=2
�
X
= 1 + (8sσ2)2
= 1 + 128s 2σ4
128s
2σ4
.
e
≤
(Jensen)
(Jensen again)
(Lemma 1.4)
for
s
|
| ≤
1
16σ2
Sub-exponential random variables also give rise ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
λ2
∧
.
t
)
λ
�
(cid:21)
−
�
(cid:20)
Proof. Without loss of generality, assume that λ = 1 (we can always replace
Xi by Xi/λ and t by t/λ. Next, using a Chernoff bound, we get for any s > 0
¯
IP( X > t)
≤
n
IE e
sXi −
e
snt
.
i=1
Y
�
(cid:2)
(cid:3)
1.4. Maximal inequalities
25
s
Next, if
|
butio... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
for the average
¯X. In many instances, we will be interested in controlling the maximum over
the parameters of such linear combinations (this is because of empirical risk
minimization). The purpose of this section is to present such results.
Maximum over a finite set
We begin by the simplest case possible: the maxi... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
2)
log IE max e sXi
i
≤
≤
IE e sXi
log
N
(cid:2)
1
(cid:3)
1
N
i
�
X
(cid:2)
≤
≤
σ2
s
e
2
2
log
(cid:3)
N
1
i
X
�
≤
≤
log N σ2s
+
s
2
Taking s =
2(log N )/σ2 yields the first inequality in expectation.
The first inequality in probability is obtained by a simple union bound:
�
IP
(
max Xi > t
1
≤
i
≤
N
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
< t) = [IP(X1 < t)]N
1
i
≤
≤
N
0 , N
.
→ ∞
→
On the opposite side of the picture, if all the Xis are equal to the same random
variable X, we have for any t > 0,
IP( max Xi < t) = IP(X1 < t) > 0
1
i
≤
≤
N
N
∀
≥
1 .
In the Gaussian case, lower bounds are also available. They illustrate the effect
of the corre... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
θ
F
{
P
,
}
∈
where P
maximum over
useful lemma.
⊂
F
IRd is a polytope with N vertices. While the family
is infinite, the
can be reduced to the a finite maximum using the following
F
Lemma 1.15. Consider a linear form x
convex polytope P
IRd ,
⊂
c x, x, c
⊤
∈
→
IRd . Then for any
max c x = max c x
x
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
the two quantities are equal.
It immediately yields the following theorem
IRd and let
IRd be a random vector such that, [v(i)]⊤X, i = 1, . . . , N are sub-Gaussian
Theorem 1.16. Let P be a polytope with N vertices v(1), . . . , v(N )
X
random variables with variance proxy σ2 . Then
∈
∈
IE[max θ⊤X]
P
θ
∈
σ
2 l... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
norm
|2 at most 1. Formally, it is defined by
u
|
IRd :
B2 = x
{
∈
d
i=1
�
X
2
x
i
≤
1 .
}
Clearly, this ball is not a polytope and yet, we can control the maximum of
random variables indexed by
B2. This is due to the fact that there exists a
finite subset of
B2 such that the maximum over this finite set is ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
N
B2.
(0, 1). Then the unit Euclidean ball
|N | ≤
Proof. Consider the following iterative construction if the ε-net. Choose x1 =
0. For any i
|2 > ε for
all j < i. If no such x exists, stop the procedure. Clearly, this will create an
ε-net. We now control its size.
2, take any xi to be any x
∈ B2 such that
xj... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
∈
IRd be a sub-Gaussian random vector with variance
IE[max θ⊤X] = IE[max
∈B2
∈B2
θ
θ
θ⊤X
]
|
≤
|
√
4σ d .
Moreover, for any δ > 0, with probability 1
δ, it holds
max θ⊤X = max
θ
∈B2
∈B2
θ
|
θ⊤X
| ≤
4σ d + 2σ
2 log(1/δ) .
�
−
√
Proof. Let
satisfies
x such that
be a 1/2-net of
N
6d . Next, observe that f... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
max θ⊤X > t
θ
∈B2
IP
≤
z
∈N
2 max z ⊤X > t
(
)
(
To conclude the proof, we find t such that
t2
8σ2
e
−
≤
≤ |N |
6d
e
−
t2
8σ2
.
)
2
t +d log(6)
e 8σ2
−
δ
≤
⇔
t2
≥
8 log(6)σ2d + 8σ2 log(1/δ) .
Therefore, it is sufficient to take t =
√
8 log(6)σ d + 2σ
2 log(1/δ) .
�
�
1.5. Problem set
30
1.5 PR... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
. . . , Zn are
freedom) if it has the same distribution as Z1
iid
(0, 1).
n
N
(a) Let Z
Z 2
∼ N
1 satisfies
−
(0, 1). Show that the moment generating function of Y =
φ(s) := E e =
sY
(cid:2)
(cid:3)
(b) Show that for all 0 < s < 1/2,
e
√
1
s
−
−
∞
2s
if s < 1/2
otherwise
(c) Conclude that
φ(s)
exp... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
�
≤
1
j
≤
≤
are iid sub-Gaussian random variables with variance proxy σ2 .
n be a random matrix such that its entries
m
{
(a) Show that the matrix A is sub-Gaussian. What is its variance proxy?
(b) Let
A
1
1
denote the operator norm of A defined by
max
IRm
x
∈
|
Ax
|2
x
|2
|
.
Show that there exits a constant ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
�
Xi| ≤
4eσ
log n .
�
Problem 1.6. Let K be a compact subset of the unit sphere of IRp that
Nε with respect to the Euclidean distance of IRp that satisfies
admits an ε-net
p are positive constants.
1 and d
|Nε| ≤
Let X
subGp(σ2) be a centered random vector.
(C/ε)d for all ε
(0, 1). Here C
≤
≥
∈
Show that th... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
ε) .
Problem 1.8. Let X1, . . . , Xn be n independent and random variables such
that IE[Xi] = µ and var(Xi)
(0, 1) and assume without loss of
generality that n can be factored into n = K
G where G = 8 log(1/δ) is a
positive integers.
σ2 . Fix δ
≤
∈
·
For g = 1, . . . , G, let X¯g denote the average over the g... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
clude.
�
6
6
6
r 2
e
t
p
a
h
C
Linear Regression Model
In this chapter, we consider the following regression model:
Yi = f (Xi) + εi,
i = 1, . . . , n ,
(2.1)
where ε = (ε1, . . . , εn)⊤ is sub-Gaussian with variance proxy σ2 and such that
IE[ε] = 0. Our goal is to estimate the function f under a li... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
tumor given certain inputs
for a new (unseen) patient.
A natural measure of performance here is the L2-risk employed in the in-
troduction:
ˆfn(Xn+1)]2
ˆR(fn) = IE[Yn+1 −
= IE[Yn+1 −
where PX denotes the marginal distribution of Xn+1. It measures how good
the prediction of Yn+1 is in average over realizations of Xn+1. ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
of these values.
In many instances, fixed design can be recognized from their structured
form. A typical example is the regular design on [0, 1], given by xi = i/n, i =
1, . . . , n. Interpolation between these points is possible under smoothness as-
sumptions.
Note that in fixed design, we observe µ∗+ε, where µ∗ =
f (x1... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
=
1
n
|
X ˆ(θ
θ∗)
|
−
ˆ
2
2 = (θ
θ
∗)⊤
−
X X
⊤
n
ˆ
(θ
−
θ∗) .
(2.2)
(2.3)
∈
A natural example of fixed design regression is image denoising. Assume
that µ∗i , i
1, . . . , n is the grayscale value of pixel i of an image. We do not
get to observe the image µ∗ but rather a noisy version of it Y = µ∗ + ε. Given
IRn, our go... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
interested in estimating Xθ∗ and not θ∗ itself, so by exten-
sion, we also call µˆls = Xθls least squares estimator. Observe that µˆls is the
projection of Y onto the column span of X.
ˆ
It is not hard to see that least squares estimators of θ∗ and µ∗ = Xθ∗ are
maximum likelihood estimators when ε
(0, σ2In).
∼ N
Propos... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
atisfies
ˆ
∼
subGn(σ2).
ˆ
IE MSE( θls) = IE
X
1
n
X
ˆ
θls
|
X
θ∗
|
−
2
2 . σ
2 r
n
,
(cid:2)
where r = rank(X⊤X
(cid:3)
r, for any δ > 0, with probability 1
). Moreove
δ, it holds
−
MSE(Xˆθ ) . σ
ls
2 r + log(1/δ)
n
.
Proof. Note that by definition
Y
|
−
Xˆθls
2
2 ≤ |
|
Y
Xθ∗
2
2 =
|
2
2 .
ε
|
|
−
(2.4)
Moreover,
Y
|
−
X... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
traditional technique is “sup-out” θls.
This is typically where maximal inequalities are needed. Here we have to be a
bit careful.
ˆ
Let Φ = [φ1, . . . , φr]
. In particular, there exists ν
∈
×
IRn
r be an orthonormal basis of the column span
ˆ
θ∗) = Φν. It yields
IRr such that X(θls
of X
ε⊤X ˆ(θls
X ˆ(θls
θ∗)
−
θ∗)
|2... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
1.19 that
−
δ, it follows from the last step in the proof1 of
sup (ε˜⊤u)2
u
∈B2
≤
8 log(6)σ
2r +
8σ2 log(1/δ) .
Remark 2.3. If d
≤
n and B := X X has rank d, then we have
⊤
n
ˆls
θ
|
θ∗
2
2 ≤
|
−
E
MS (Xˆθls)
λmin(B)
,
and we can use Theorem 2.2 to bound
ˆ
θls
θ∗
2
2 directly.
|
1we could use Theorem 1.19 directly here... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
��v)
K
K
−
v
∈
XK
θ
∈
K
IRn. This is a measure of the size (width) of
where XK =
Xθ : θ
XK. If ε
(0, Id), the expected value of the above supremum is actually
called Gaussian width of XK. Here, ε is not Gaussian but sub-Gaussian and
similar properties will hold.
{
∼ N
} ⊂
∈
ℓ1 constrained least squares
Assume here that... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
Theorem 2.4. Let K =
θ∗
of X are normalized in such a way that maxj |
ˆ
least squares estimator θls
B1 satisfies
1
= IE
n
2X
θ∗
|2 . σ
r
log d
n
,
ˆ
Xθls
B1 −
MSE(Xθls )
ˆ
B
IE
≥
|
1
(cid:2)
(cid:3)
2.2. Least squares estimators
39
Moreover, for any δ > 0, with probability 1
MSE(Xˆθls
B1 ) . σ
r
δ, it holds
−
log(d/δ)
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
�ls
B
1 benefits from the best of both rates.
ˆθls
B
1 (exercise!) so that
ˆ
ls
IE MSE(Xθ 1 ) . min
B
r
n
,
r
log d
n
.
(cid:17)
(cid:16)
round r √
This is called an elbow effect. The elbow t
akes place a
logarithmic terms).
≃
(cid:3)
(cid:2)
n (up to
ℓ0 constrained least squares
We abusively call ℓ0 norm of a vector θ
I... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
θls
K when K =
B0(k). Note that
computing θls
least squares estimators,
which is an exponential number in k. In practice this will be hard (or even
(cid:1)
(cid:0)
impossible) but it is interesting to understand the
tatistical properties of this
s
estimator and to use them as a benchmark.
essentially requires computing... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
) so that θls
∈ B0(2k). For
θ∗
K −
submatrix of X that is obtained
S
× |
the rank of XS and
S of X. Denote by rS ≤ |
S
rS be an orthonormal basis of the column span
S
| to be the vector with
IR|
ˆ
S. If we denote by S = supp(θls
2k
K −
IRrSˆ such that
IRd, define θ(S)
ˆ
θ∗), we have
| ≤
ˆ
S
∈
∈
∈
×
|
|
|
∈
X ˆ(θls
K
θ∗)... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
2
= k
SX|
|
u
∈
(cid:0)
r
B2
S
(cid:1)
It follows from the proof of Theorem 1.19 that for any
S
2k,
sup (ε˜⊤u)2 > t
u
rS
2
∈B
IP
(cid:0)
S
6|
|e−
t
28σ
≤
≤
(cid:1)
|
62ke−
| ≤
28σ .
t
2.2. Least squares estimators
Together, the above three displays yield
IP(
|
Xˆ
θls
XK −
θ∗
2
|2 > 4t)
≤
62ke− 28σ .
t
d
2k
(cid:18) (c... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
k
k
−
k + 1
=
eknk+1
(k + 1)k+1
1 +
1 k
k
,
(cid:18)
(cid:19)
(cid:18)
(cid:19)
(cid:17)
where we used the induction hypothesis in the first inequality. To conclude, it
suffices to observe that
(cid:16)
(cid:16)
(cid:17)
1 +
(cid:16)
k
1
k
(cid:17)
e
≤
It immediately leads to the following corollary:
Corollary 2.8. Under ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
)
∞
0
Z
H +
H +
=
≤
≤
= H +
∞
0
Z
2k
d
j
IP(
|
Xˆθls
K −
Xθ∗
|
2
2 > nu)du
IP(
|
Xˆθls
K −
Xθ∗
|
2
2 > n(u + H))du
62k
∞
n(u+H)
32σ
2
,
e
−
(cid:19)
0
Z
62
k
e
−
nH
2
32σ
2k
j=1
X (cid:18)
d
j
j=1 (cid:18) (cid:19)
X
32σ2
n
du .
Next, take H to be such that
2k
j=1
X
d
j
(cid:18)
62ke−
nH
2
32σ = 1 .
(cid:19)
H .
σ2k
n
... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
to
in this sequence model and we will also discuss this case. Its links
to nonparametric estimation will become clearer in Chapter 3. The goal here
is to estimate the unknown vector θ∗.
∞
N
The sub-Gaussian Sequence Model
Note first that the model (2.7) is a special case of the linear model with fixed
design (2.1) with n... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
�
ˆ
θ) = (θ
θ∗)⊤
−
X⊤X
n
ˆ
(θ
−
θ∗) =
ˆ
θ
|
−
θ∗
2
2 .
|
2.3. The Gaussian Sequence Model
44
Furthermore, for any θ
∈
IRd, the assumption ORT yields,
y
|
θ
|
−
2
2 =
1
n
|
X⊤Y
θ
2
2
|
−
2
2 −
Xθ
2
n
1
θ⊤X⊤Y + Y ⊤XX⊤Y
n2
2
2 −
2
n
(Xθ)⊤Y +
1
n
Y 2
|2 + Q
|
|
=
=
=
θ
|
|
1
n
1
n |
|
Y
Xθ
|
−
2
2 + Q ,
(2.8)
where Q is a... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/619e4ae252f1b26cbe0f7a29d5932978_MIT18_S997S15_CourseNotes.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.