text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
π
2
∇ (Xθ) · Xθ
F
(cid:48) (cid:17)
dθ,
If we condition on Xθ, since (cid:13)
with variance at most (cid:0)λ π
2 ∇F (Xθ)(cid:13)
(cid:13)λ π
2 σ, λ π
2 σ(cid:1)2. This directly implies that, for every value of Xθ
2 ∇F (Xθ) · X (cid:48)
(cid:13) ≤ λ π
θ is a gaussian random variable
EX (cid:48)
θ
exp
(cid:16)
λ
π
2
∇F (... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
independent (standard) gaussian random variables and it is easy to see that it is a
function of these variables, since
2-Lipschitz
√
(cid:12)
(cid:12)(cid:107)W (1)(cid:107) − (cid:107)W (2)(cid:107) ≤ W (1) − W (2) ≤ W (1) − W (2)
(cid:12)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
4
=
1
n
.
4.2.2 Talagrand’s concentration inequality
A remarkable result by Talagrand [Tal95], Talangrad’s concentration inequality, provides an analogue
of Gaussian concentration to bounded random variables.
Theorem 4.9 (Talangrand concentration inequality, Theorem 2.1.13 [Tao12]) Let K > 0,
≤ i ≤ n. Let
and let X1, .... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
p−ε(cid:35)n
p + ε
1 − p − ε
(cid:41)
Xi ≤ p − ε
≤
(cid:34)(cid:18) p
p − ε
(cid:19)p−ε (cid:18) 1 − p
1
− p + ε
(cid:19)1− +ε
p
(cid:35)
n
4.3.2 Multiplicative Chernoff Bound
There is also a multiplicative version (see, for example Lemma 2.3.3. in [Dur06]), which is particularly
useful.
Theorem 4.11 Let X1, . . . , Xn ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
} ≤ exp(−x),
(cid:107) (cid:107)2 (cid:80)n
where a 2 = k=1 a2
k and (cid:107)a(cid:107) = max
∞
1≤k
≤n | k|
a .
Note that if ak = 1, for all k, then Z is a χ2 with n degrees of freedom, so this theorem immediately
gives a deviation inequality for χ2 random variables.
4.4 Matrix Concentration
In many important applicat... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
derivation of Proposition 4.8, that allowed us to easily transform
bounds on the expected spectral norm of a random matrix into tail bounds, we will mostly focus on
bounding the expected spectral norm. Tropp’s monograph [Tro15b] is a nice introduction to matrix
concentration and includes a proof of Theorem 4.13 as well... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)
n
(cid:88)
(gk − hk)2
(cid:118)
(cid:117)
(cid:117)
(cid:116)
n
(cid:88)
k=1
(vT Akv)2
2
(vT Akv)
(cid:107)g − h(cid:107)2,
where the first inequality made use of the triangular inequality and the last one of the Cauchy-Schwarz
inequality.
This motivates us to define a new parameter, the weak variance σ .∗
Definition 4.... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)
(cid:13)
(cid:13)
n
(cid:88)
k=1
(cid:13)
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
A2
k
and σ∗ =
(cid:118)
(cid:117)
(cid:117)
(cid:116)
max
v: (cid:107)v(cid:107)
=1
n
(cid:88)
k=1
2
(v Akv) .
T
We have
σ ≤ σ.
∗
Proof. Using the Cauchy-Schwarz inequality,
σ2 = max
∗
v: (cid:107)v(cid:107)=1
= max
v: (cid:107)v(cid:107)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:13)
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
= E
(cid:32)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
k=1 gkAk(cid:107).
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
kA
(cid:33)
2
g
k
n
(cid:88)
k=1
= E max v
v: (cid:107)v(cid:107)=1
(cid:33)
2
gkAk
v
(cid:32)
n
(cid:88)
T
k=1
≥ max EvT
v: (cid:107... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:16) d, meaning that the bound given by NCK (Theorem 4.14) is, in this
The next example will show that the logarithmic factor is in fact needed in some examples
for k = 1, . . . , d. The matrix (cid:80)
n
Example 4.18 Consider Ak = ekek
k=1 gkAk corresponds to
a diagonal matrix with independent standard gaussian random... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
and σ2
ij
∗ (cid:16) b2
ij.
This means that, when the random matrix in NCK (Theorem 4.14) has negative entries (modulo
symmetry) then
E(cid:107)X(cid:107) (cid:46) σ + (cid:112)log dσ∗.
(39)
18By a (cid:16) b we mean a (cid:46) b and a (cid:38) b.
68 Theorem 4.19 together with a recent improvement of Theorem 4.14 by Tr... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
the
largest variance is comparable to the variance of a sufficient number of entries, then the bound in
Theorem 4.19 is tight up to constants.
However, the situation is not as well understood when the variance profiles
Since the spectral norm of a matrix is always at least the (cid:96)2 norm of a row, the
holds (for X a s... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
of Theorem 4.14)
Theorem 4.22 Let H1, . . . , Hn ∈ Rd×d be symmetric matrices and ε1, . . . , εn i.i.d. Rademacher
random variables (meaning = +1 with probability 1/2 and = −1 with probability 1/2), then:
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
1 + 2(cid:100)log(d)(cid:101)
εkHk ≤
(cid:17) 2 σ,
(cid:13)
n
(cid:13)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
4.23.
Note that, when the matrices Hk are diagonal, this problem corresponds to Spencer’s Six Standard
Deviations Suffice Theorem [Spe85].
Remark 4.24 Also, using Theorem 4.22, it is easy to show that if one picks εi as i.i.d. Rademacher
random variables, then with positive probability (via the probabilistic method) the ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
70 4.6.2 Back to matrix concentration
Using Theorem 4.22, we’ll prove the following Theorem.
Theorem 4.26 Let T1, . . . , Tn ∈ Rd×d be random independent positive semidefinite matrices, then
E
(cid:13)
n
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)
i=1
Ti
where
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13) ≤ (cid:13)
(c... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:13)
(cid:13)
(cid:13)
i=1
εiTi
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
,
71 Proof. [of Theorem 4.26]
Using Lemma 4.27 and Theorem 4.22 we get
E
(cid:13)
n
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)
i=1
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
Ti
≤
(cid:13)
n
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)
i=1
E
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:13)
(cid:13)
(cid:13)
(cid:13) ≤ (cid:13)
(cid:13)
(cid:13)
(cid:13)
n
(cid:88)
i=1
ETi
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
+ (cid:112)C(d)E
(cid:18)
max
i
(cid:107)Ti(cid:107)
(cid:19) 1
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
n
(cid:88)
i=1
1
2
.
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Y 2
i
1
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:32)
≤
E
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
n
(cid:88)
i=1
Y 2
i
(cid:33) 1
2
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
,
and the proof can be concluded by noting that Y 2
i (cid:23) 0 and using Theorem 4.26.
(cid:50)
Remark 4.29 (The rectangular c... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
∈ Rd×d and non-negative integers r, q satisfying
q ≤ 2r,
Tr (cid:2)HW qHY 2r−q(cid:3) + Tr (cid:2)HW 2r−qHY q(cid:3) ≤ Tr (cid:2)H 2 (cid:0)W 2r + Y 2r(cid:1)(cid:3) ,
and summing over q gives
2r
(cid:88)
q=0
(cid:2)
Tr HW qHY 2 −q
r
(cid:18)
(cid:3) ≤
2r +
2
(cid:19)
1
Tr (cid:2)H 2 (cid:0)W 2r
Y 2r(cid:1)(cid:3)
+
We... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
2p. We introduce X+i and X−i as X conditioned on εi being, respectively +1 or
−1. More precisely
X+i = Hi +
εjHj and X−i = −Hi +
εjHj.
(cid:88)
(cid:88)
Then, we have
j(cid:54)=i
i
j=(cid:54)
E Tr X 2p = E Tr (cid:2)XX 2p−1(cid:3)
n
= E (cid:88)
Tr εiHiX 2p−1.
Note that E Tr (cid:2)ε H X 2p−1(cid:3) = 1
i
i
εi
2 Tr
(ci... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
34)(cid:32)
n
(cid:88)
i=1
(cid:33)
(cid:35)
H 2
i
X 2p−2
≤
(cid:13)
n
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)
i=1
H 2
i
(cid:13)
(cid:13)
(cid:13)
(cid:13) Tr X
(cid:13)
2p
−
2
2
= σ Tr X
− ,
2p 2
which gives
Applying this inequality, recursively, we get
E Tr X 2p
≤ σ2(2p − 1)E Tr X 2p−2.
E Tr X 2p ≤ [(2p − 1)(2p ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
32 Note that, up until the step from (44) to (45) all steps are equalities suggesting that
this step may be the lossy step responsible by the suboptimal dimensional factor in several cases (al-
though (46) can also potentially be lossy, it is not uncommon that H 2
is a multiple of the identity
i
matrix, which would ren... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
ve or disprove: there exist positive universal constants c1 and c2 such that
For any U ∈ Rn×d for which U T U = Id×d
Prob
(cid:8)(cid:13)
(ΠU )T (ΠU ) − I ≥ ε < δ,
(cid:13)
(cid:13)
(cid:13)
(cid:9)
for m
≥ c
1
d+log( 1
δ )
ε2
and s ≥ c2
log( d
δ )
ε2
.
2. Same setting as in (1) but conditioning on
m
(cid:88)
r=1
δri =... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
T
constants such that
k = 1 I
m d×d
. The conjecture is then that, there exists c1 and c2 positive universal
Prob
(cid:40)(cid:13)
m
(cid:13)
(cid:13)(cid:88)
(cid:13)
(cid:13)
k=1
(cid:2)zkzT
k − Ezkz
T
k
(cid:41)
≥ ε < δ,
(cid:13)
(cid:13)
(cid:3)(cid:13)
(cid:13)
(cid:13)
for m ≥
c1
d+log( 1
δ )
ε2
and s ≥ c2
log( d... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
corresponds to the Kronecker product. Note that
EA⊗k = A ⊗
(cid:19)
J
,
(cid:18)
1
k
where J = 11T is the all-ones matrix.
77 Open Problem 4.5 (Random k-lifts of graphs) Give a tight upperbound to
E
(cid:13)
(cid:13)
(cid:13)A⊗k − EA⊗k(cid:13)
(cid:13)
(cid:13) .
Oliveira [Oli10] gives a bound that is essentially of th... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, X = {x1, . . . , xn}, in Rd (with d large). If d > n, since the points have
to lie in a subspace of dimension n it is clear that one can consider the projection f : Rd → Rn of
the points to that subspace without distorting the geometry of X. In particular, for every xi and xj,
(cid:107)f (xi) − f (xj)(cid:107)2 = (ci... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
.i.d standard Gaussian random variables and Y =
(y , . . . , y ). Let g : Rd
=
(cid:107)Y (cid:107) (y1, . . . , yk) and L = (cid:107)Z(cid:107)2. It is clear that EL = k . In fact, L is very concentrated around its
mean
→ R be the projection into the first k coordinates and Z = g
(cid:16) Y
(cid:107)Y (cid:107)
(cid:17... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)2
(cid:107)xi − xj(cid:107)2
(cid:21)
(cid:32)
≤ (1 − (cid:15))
≤ exp
−
(cid:33)
k (cid:0)(cid:15)2 − 2(cid:15)3/3(cid:1)
4
≤ exp (−2 log n) =
1
n2
.
By union bound it follows that
2
n2
Since there exist (cid:0)n(cid:1) such pairs, again, a simple union bound gives
(cid:20) (cid:107)f (xi) − f (xj)(cid:107)2
(cid:107)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
3
log n.
Then, for any set X of n points in Rd, take f : Rd → Rk to be a suitably scaled projection on a random
subspace of dimension k, then f is an (cid:15)−isometry for X (see (48)) with probability at least 1 − 1 .nτ
Lemma 5.3 is quite remarkable. Think about the situation where we are given a high-dimensional
data... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
intuition, there is a lot of hand-
waving!!)
Let’s continue thinking about the high-dimensional streaming data. After we draw the random
projection matrix, say M , for each data point x, we still have to compute M x which, since M has
O((cid:15)−2 log(n)d) entries, has a computational cost of O((cid:15)−2 log(n)d). In ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
in some sense, the motivation for Open Problem 4.4.
We recommend these notes Jelani Nelson’s notes for more on the topic [Nel].
5.2 Gordon’s Theorem
In the last section we showed that, in order to approximately preserve the distances (up to 1 ± ε)
between n points it suffices to randomly project them to Θ (cid:0)(cid:15)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
).
Then
√
(cid:113) k
k+1
k ≤ ak ≤
√
k.
We are now ready to present Gordon’s Theorem.
Theorem 5.6 (Gordon’s Theorem [Gor88]) Let G ∈ Rk×d a random matrix with independent
N (0, 1) entries and S ⊂ Sd−1 be a closed subset of the unit sphere in d dimensions. Then
E max
x∈S
(cid:13)
1
(cid:13)
(cid:13)
(cid:13) ak
Gx
(cid:... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
itz. Thus, one can use Gaussian
ak
(cid:107)(G1 − G2) x(cid:107)
x∈S
(cid:107)| ≤
≤ max |(cid:107)G1x(cid:107) − (cid:107)G2x
max
x∈S
= (cid:107)G1 − G2(cid:107) ≤ (cid:107)G1 − G2(cid:107)F .
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
Concentration to get:
and
(cid:26)
(cid:27)
(cid:18)
Prob
max (cid:107)Gx... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
ak
, using (49), (50), and recalling that a2
k ≤ k. (cid:50)
Remark 5.8 Note that a simple use of a union bound23 shows that ω(S) (cid:46) (cid:112)2 log |S|, which means
that taking k to be of the order of log |S| suffices to ensure that 1
G to have the Johnson Lindenstrauss
ak
property. This observation shows that Theo... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Theorem
In order to prove this Theorem we will use extensions of the Slepian’s Comparison Lemma.
Slepian’s Comparison Lemma, and the closely related Sudakov-Fernique inequality, are crucial
tools to compare Gaussian Processes. A Gaussian process is a family of gaussian random variables
indexed by some set T , {Xt}t T (... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Xt1,u2] ≤ E
2
[Yt1,u1 − Yt1,u2]
,
(52)
24Although intuitive in some sense, this turns out to be a delicate statement about Gaussian random variables, as it
does not hold in general for other distributions.
84 and, for t1 (cid:54)= t2,
then
E [Xt1,u1 −
Xt2,u2] ≥ E [Yt1,u1 − Yt2,u2]
2
2
,
(53)
E
(cid:20)
min max Xt,u
t∈T... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
v = E max gT u E max( hT v) = ak
−
−
−
ω(S),
u∈Sk−1
v∈S
u
∈Sk−1
v∈S
gives the second part of the Theorem.
On the other hand, since E (cid:12)
(cid:12)Av,u − Av(cid:48),u(cid:48)
Theorem 5.10 with X = B and Y = A, to get
(cid:12)
(cid:12)
2 − E (cid:12)
(cid:12)Bv,u − Bv(cid:48),u(cid:48)
(cid:12)
(cid:12)
2 ≥ 0 then we... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
if we know more about the structure of x, a prime example is
when x is known to have few non-zero entries (being sparse). Sparse signals do arise in countless
applications (for example, images are known to be sparse in the Wavelet basis; in fact this is the basis
of the JPEG2000 compression method).
We’ll revisit spars... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
∈ SN −1, (cid:107)x(cid:107)0 ≤ k is the set of 2s
sparse vectors, and ω (S2s) the gaussian width of S2s.
(cid:9)
(cid:8)
Proposition 5.12 If s ≤ N , the Gaussian Width ω (Ss) of Ss, the set of unit-norm vectors that are
at most s sparse, satisfies
ω (Ss) (cid:46) s log
(cid:18)
(cid:19)
N
s
.
Proof.
ω (Ss) =
max
v∈SSN ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:19)(cid:21)
.
(cid:18) N
s
(54)
Taking u > log (cid:0) N
s
(cid:1) it can be readily seen that the typical size of maxΓ⊂[N ], |Γ|=s (cid:107)gΓ(cid:107)2 is (cid:46)
(cid:1) The proof can be finished by integrating (54) in order to get a bound of the expectation of
.
s log (cid:0) N
s
(cid:113)
maxΓ⊂[N ], |Γ|=s (cid:10... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:0)s, 1
3
(cid:18) N
s
(cid:1)-RIP, with high probability.
M ≥ Cs log
(cid:19)
,
Theorem 5.14 suggests that RIP matrices are abundant for s ≈ M
log(N ) , however it appears to be
very difficult to deterministically construct matrices that are RIP for s (cid:29)
M , known as the square
bottleneck [Tao07, BFMW13, BFMM1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
high probability, certify that a gaussian matrix A does not have s-sparse
vectors in its nullspace?
3
The second√ question is tightly connected to the question of sparsest vector on a subspace (for
which s ≈ M is the best known answer), we refer the reader to [SWW12, QSW14, BKS13b] for
more on this problem and recent a... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
0)(cid:8)X : X ∈ Rn1×n2, rank(X) ≤ r(cid:9)(cid:1) = E max Tr(GX).
(cid:107)
X(cid:107)F
rank(
=1
X)≤r
Let X = U ΣV T be the SVD decomposition of X, then
ω (cid:0)(cid:8)X : X ∈ Rn1×n2, rank(X) ≤ r(cid:9)(cid:1) = E
max
U T U =V T V =Ir×r
Σ∈Rr×r diagonal (cid:107)Σ(cid:107)F =1
Tr(Σ V T GU ).
(cid:1)
(cid:0)
This impli... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Let us begin with some motivating examples. Let Pn denote the path on n vertices. Each Pn
naturally embeds into the bi-infinite path whose vertices are the set of integers Z with edges between
consecutive integers. By an abuse of notation we denote the bi-infiite path as Z. It is intuitive to
say that Pn converges to Z a... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
common path (see the red vertices in Figure 1). Pick exactly
one vertex from each cluster, say the median vertex. Connect each such vertex, say (n, y), to the
2
− 1, y), which is the unique edge connecting (n, y) to
vertex (n − 1, y) via the edge (n, y) ↔ (n
Z[−n + 1, n − 1] . If a vertex (n, y(cid:48)) on the right ha... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
We will in fact calculate the tree entropy.
−
The tree entropy of a sequence of bounded degree connected graphs {Gn} is the limiting value of
log sptr(Gn)/|Gn| provided it exists. It measures the exponential rate of growth of the number of
spanning trees in Gn. We will see that that tree entropy exists whenever the gra... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
G, x) ∈ G :
Nr(G, x) ∼= Nr(H, y)}. A random rooted graph (G, ◦) is a probability space (G, F , µ); we think of
(G, ◦) as a G-valued random variable such that P(cid:2) (G, ◦) ∈ A (cid:3) = µ(A) for every A ∈ F .
Let us see some examples. Suppose G is a finite connected graph of maximum degree ∆. If ◦G is a
uniform random... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, y)|
/|Gn| converges as n → ∞. Using tools from measure theory
G, ◦) ∈ G such that
that there is a random rooted graph (
and compactness of G it can be shown
(cid:3)
if the ratios in the previous sentence converge then P Nr(Gn, ◦n) =∼ Nr(G, ◦)
in distribution to (G, ◦
every r. This is what it means for (Gn, ◦n) to con... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, X
SRW is a Markov process given by the transition matrix P (u, v) = 1u∼v where u
∼ v means that
{u, v} is an edge of G. If G has bounded degree then if is easily verified that P(cid:2) Xk = y | X0 = x =
P k(x, y). The k-step return probability to x is pk
G(x) = P k(x, x) for k ≥ 0.
deg(u)
−
(cid:3)
Suppose that Gn is ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
representation theorem. On the event {Nk/2(G(cid:48)
pk
G ((cid:48)
G ((cid:48) ◦). Therefore,
◦n) = pk
n, ◦(cid:48)
n, ◦(cid:48)
n, ◦(cid:48)
n
(cid:12)E(cid:2) pk (◦ ) (cid:3) − E(cid:2) pk (◦) (cid:3)(cid:12)
(cid:12)
n
G
Gn
(cid:48) )
k
◦n − G(cid:48) ◦
p
( (cid:48))
(cid:12)
(cid:3)
(cid:12)
(cid:48)
n
G
(cid:12)E... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
First, we would have to calculate the number of d-regular graphs on n vertices. This is no easy
task. To get around this issue we will consider a method for sampling (or generating) a random
d-regular multigraph (that is, graphs with self loops and multiple edges between vertices), This
sampling procedure is simple eno... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
means that for every r > 0 we must show that
(cid:104) |v ∈ V (Gn,d) : Nr(Gn,d, v) ∼= Nr(Td, ◦)|
n
(1)
E
(cid:105)
→ 1 as n → ∞,
where ◦ is any fixed vertex of Td (note that Td is vertex transitive). Notice that if Nr(Gn,d, v)
contains no cycles then it must be isomorphic to Nr(Td, ◦) due to Gn,d being d-regular. Now
su... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, E C(cid:96) ≤ (3d − 3)(cid:96) if d ≥ 3.
(cid:2)
C(cid:96) be the number of cycles of length (cid:96) in Gn,d. Then limn
→∞
(cid:2)
E C
(cid:3)
(cid:96) =
(
1)(cid:96)
d−
2(cid:96)
.
Proof. Given a set of (cid:96) distinct vertices {v1, . . . , v(cid:96)} the number of ways to arrange them in cyclic
order is ((cid:96... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
= (1 + o(1))(nd)−(cid:96) as n → ∞ and it is at most 3(cid:96)(nd)−(cid:96) if d ≥ 3 (provided
(cid:96)!
(cid:96)
(cid:96)
(cid:96)
94 that (cid:96) is fixed).
(3d − 3)(cid:96) if d ≥ 3.
It follows from these observations that E(cid:2) C(cid:96)
(cid:3) → (d − 1)(cid:96)/(2(cid:96)), and is at most
MUSTAZEE RAHMAN
3. En... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
for the tree entropy of Zd and Td and asymptotically enumerate the number of spanning trees in
the grid graphs Z[−n, n]d and random regular graphs Gn,d.
3.1. The Matrix-Tree Theorem. Let G be a finite graph. Let D be the diagonal matrix consisting
of the degrees of the vertices of G. The Laplacian of G is the |G| × |G| ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
5) of exercise 2.2 we see that the Laplacian L of
G has n = |G| eigenvalues 0 = λ0 < λ1 ≤ · · · ≤ λn−1. The Matrix-Tree Theorem states that
(2)
sptr(G) =
1
n
n−1
(cid:89)
i
=1
λi.
In other words, the number of spanning trees in G is the product of the non-zero eigenvalues of
the Laplacian of G. In fact, the Matrix-Tree... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
values. Perhaps the easiest way to see this is to define a new inner
(cid:80)
product on RV (G) by (f, g)π = x∈V (G) π(x)f (x)g(x) where π(x) = deg(x)/2e and e is the number
called the stationary measure of the SRW on G. It is a probability
of edges in G. The vector π is
(cid:80)
distribution on V (G), that is,
x π(x) =... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
reader.
From 3 and |G| being n we deduce that
log sptr(G)
|G|
=
log 2e(G)
n
+
(cid:80)
x V
∈ (G)
log deg(x) (cid:80)n−1
n
+ i=1 log(1 − µi)
n
.
96 MUSTAZEE RAHMAN
(cid:80)
Since
(cid:80)
k 1 xk/k for −1 ≤
≥
log(1−x) = −
−1
n
i = TrP k
i=1 µk
Now,
− 1 since we exclude the eigenvalue 1
Note that TrP k = (cid:80)
x V (G) ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
deterministic, vertex transitive graph of degree d. If ◦ ∈ V (G) is
any fixed vertex then the tree entropy of G is defined to be
h(G) = log d −
(cid:88) 1
k≥1
pk
k G(
◦).
The tree entropy h(G) does not depend on the choice of the sequence of graphs Gn converging to G
in the local weak limit.
To prove this theorem let ◦n ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�n) − |G
for some α > 0.
G (
Gn
−α
≤
n)
◦
|
|
n
Then it follows from the dominated
≥1 pk
(cid:3)
G(◦)/k ,
verges to E
(cid:2) (cid:80)
k
as required.
con
vergence
theorem that (cid:80) 1
k
k≥1
(E(cid:2) pk
G (◦
n
n)
(cid:3) − |Gn
1
|− ) con-
Lemma 3.4.
G(x) denote the k-step
return probability of the SRW on G starting ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
pose f ∈ RV (G) takes both positive and negative values. Suppose that |f (x0)| = ||f || =∞
0) ≥ 0. Let z be
t = z in G from x0
maxx V (G) |f (x)|, and by replacing f with −f if necessary we may assume that f (x
such that f (z) ≤ 0. Then ||f || ≤ |f (x0) − f (z)|. There is a path x0, x1, . . . , x
to z. Therefore,
∞
||f... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
terms (cid:112)K(x, y)|f (x)−f (y)| and (cid:112)K(x, y)(|f (x)|+
|f (y)|) then we deduce that
||f ||4
∞ ≤ e2(cid:0)(I − P )f, f (cid:1)
π · (cid:0)(I + P )|f |, |f |
(cid:1)
.π
Notice that
(cid:0)
|
(I + P )|f |, |f | ≤ 2( f
m
(P f, f )π ≤ (f, f )π because all eigenvalues of P lies in
(cid:1)
|, |f |)π. If (f, f )π ≤ ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
√
this to the function f (y) =
√1x(y)−π(x)
π(x)(1−π(x))
|| ≤ 2e(k + 1)−1/4 if f ∈ U and (f, f )π ≤ 1. Let us now apply
∞
| ≤ 2e(k + 1)−1/4. The value of P kf (x) is
. Then |P kf (x)
√
P k(x, x)(1 −
π(x)) − π(x) (cid:80)
(cid:112)
π(x)(1 − π(x))
y=x P k(y, x)
P k(x, x)(1 −
(cid:112)
=
π(x)) − π(x)(1 P
− k(x, x))
π(x)(1 ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
12)
(cid:12) (cid:88)
= |G|−1(cid:12)
(cid:12)
(cid:0)pk
(cid:12)
(cid:1)
(cid:12)
G(x) − π(x)
(cid:12)
x∈
V (G)
≤ |G|−
x∈V (G)
pk
1 (cid:88) (cid:12)
π(x) G(x)
(cid:12)
(cid:12)
π(x)
x∈V (G)
− 1
(cid:12)
(cid:12)
(cid:12)
This proves that (cid:12)
(cid:12)E(cid:2) pk (◦ ) (cid:3) − |G |−1
n
n
G
n
≤
∆
(k + 1)1/4
.
(cid... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
−
d −
2k 1
k+1
2k
( )
k
k+1
ess [9
] Lemma 1.24):
. The numbers
(◦) = k
k 1
(cid:8)
(cid:9)
2k
F (t) =
2(d
d − 1 + (cid:112)
− 1)
.
d2 − 4(d − 1)t2
k 1 pk (◦) = (cid:82) 1 F (t)−1 dt. It turns out that this integrand has an antiderivative and
≥ Td
t
0
Note that (cid:80)
it can be shown that
h(Td) = log
(cid:104)
(d
2
−... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:17)
.
99 LOCAL CONVERGENCE OF GRAPHS AND ENUMERATION OF SPANNING TREES
In the above, (·, ·) is the inner product on (cid:96)2(Zd) and 1o is the indicator function of the origin.
The inner product above may be calculated via the Fourier transform. The Fourier transform
(cid:96)2(Zd)
states that (cid:96)2(Zd) is iso... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
��ned on isomorphism classes
of such graphs, so F (G, x, y) = F (H, u, v) if φ : G → H is a graph isomorphism satisfying φ(x) = u
and φ(y) = v.
A random rooted graph (G, ◦) is unimodular if for all F as above the following equality holds
(cid:104) (cid:88)
E
F (◦, x)
(cid:105)
=
(cid:104) (cid:88)
E
F (x, ◦)
(cid:105)
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
p > 1/(d − 1). In fact,
∞(◦) is unimodular. What is the tree entropy of Cp
Cp
∞(◦)? Is it strictly increasing in p?
100 MUSTAZEE RAHMAN
References
[1] D. J. Aldous and R. Lyons, Processes on unimodular networks, Electron. J. Probab. 12 #54 (2007), pp. 1454–
1508.
[2] E. A. Bender and E. R. Canfield, The asymptotic numbe... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
as oppose to file types that save the pixel value of each pixel in the image). The idea
behind these compression methods is to exploit known structure in the images; although our cameras
will record the pixel value (even three values in RGB) for each pixel, it is clear that most collections of
pixel values will not corr... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
�
A
.
x
Last lecture
we used Gordon’s theorem to show that, if w
ok random measurements, on the
(cid:1) measurements suffice to have all considerably different s-sparse signals correspond
(cid:0)
order of s log N
s
to considerably different sets of measurements. This suggests tha... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
hits the affine subspace of interest, this
image conveys how this norm promotes sparsity, due to the pointy corners on sparse vectors.
This motivates one to consider the following optimization problem (surrogate to (55)):
min (cid:107)z(cid:107)1
s.t. Az = y,
(56)
In order for (56) we need two things, for the solution of... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
duality in Linear Programming with a game theoretic view point. The
idea will be to reformulate (57) without constraints, by adding a dual player that wants to maximize
the objective and would exploit a deviation from the original constraints (by, for example, giving the
dual player a variable u and adding to to the ob... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
v−≥0
−
T
(cid:1)
v+ AT u ω+ + 1
(cid:0) −
−
T
(cid:1)
v− + AT u ω− + uT y
(60)
With this formulation, it becomes clear that the dual players needs to set 1 − v+ − AT u = 0,
1 − v− + AT u = 0 and thus (60) is equivalent to
uT y
max
u
v+ 0
≥
−≥0
v
1−v+−AT u=0
1−v−+AT u=0
or equivalently,
maxu uT y
s.t. −1 ≤ AT
u ≤ 1.
(61... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
From (62) it is clear that u must satisfy (cid:0)1T − uT A ω+ = 0 and 1T + uT A ω− = 0, this is known
be +1 or −1 when x
as complementary slackness. This means that we must
is non-zero (and be +1 when it is positive and −1 when it is negative), in other words
(cid:1)
(cid:0)
tries of AT u
have the en
(cid:1)
(cid:0)AT ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
sign (xS) ,
where (cid:0)AT (cid:1)† = A (cid:0)
S
Corollary.
S
−
1
(cid:1)
AT
S AS
is the Moore Penrose pseudo-inverse of AT
S . This gives the following
Corollary 6.3 Consider the problem of sparse recovery discussed this lecture. Let S = supp(x), if
AS is injective and
(cid:16)
(cid:13)
(cid:13)
(cid:13)
AT
ScAS
(ci... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
3 , where
(cid:107)x(cid:107)=1 (cid:107)
2
1 + 1
(cid:107)
Bx .
(cid:1) entries then, for any |S ≤ s| we have
(cid:107) · (cid:107) denotes the operator norm (cid:107)B(cid:107) = max
This means that, if we take A random with i.i.d. N (cid:0)0, 1
M
(cid:1)−1 (cid:107) ≤ (cid:0)1 − 1
3
3 and l (cid:0)AT
S AS
(cid:113)
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
(cid:0)AT
S AS
(cid:1)−1
sign (xS)
(cid:17)
≥ 1
(cid:13)
(cid:13)
(cid:13)∞
≤ 2N exp
−
(cid:16) √
M√
3s
2
(cid:17)2
= exp
(cid:18)
−
(cid:20) M
s
3
1
2
(cid:21)(cid:19)
− 2 log(2N )
,
which means that we expect to exactly recover x via (cid:96)1 minimization when M (cid:29) s log(N ), similarly
to what was p... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
nullspace that are this concentrated on such few
entries. This motivates the following definition.
Definition 6.5 (Null Space Property) A is said to satisfy the s-Null Space Property if, for all v
in ker(A) (the nullspace of A) and all |S| = s we have
(cid:107)vS(cid:107)1 < (cid:107)vSc(cid:107)1.
From the argument abov... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
more recently, Bourgain [Bou14] achieved M = Ωδ(K log3 N ); Nelson, Price and Wootters [NPW14]
also achieved M = Ωδ(K log3 N ), but using a slightly different measurement matrix. The latest result
is due to Haviv and Regev [HR] giving an upper bound of M = Ωδ(K log2 k log N ). As far as lower
bounds, in [BLM15] it was s... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
T x
max
x
≤ δ,
or equivalently
a
(cid:13) ≤ δ.
S AS − I(cid:13)
means that AT
columns
(cid:13)
AT
(cid:13)
(cid:13)
(cid:13)AT
If the columns of A are unit-norm vectors (in RM ), then the diagonal of AT
S AS − I consists only of the non-diagonal elements of AT
aj, of A we have (cid:12)
i,
(cid:13)
S AS is all-ones, thi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
real valued, but all
of the results above hold for either case.
Consider the following 2d vectors: the d vectors from the identity basis and the d orthonormal
vectors corresponding to columns of the Discrete Fourier Transform F . Since these basis are both
orthonormal the vectors in question are unit-norm and within th... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
)|2 = α(cid:107)x(cid:107)2,
∀x∈CM ,
N
(cid:88)
k=1
vkvT
k = αI.
The smallest coherence of a set of N unit-norm vectors in M dimensions is bounded below by the
Welch bound (see, for example, [BFMW13]) which reads:
(cid:115)
µ ≥
N − M
M (N − 1)
.
One can check that, due to this limitation, deterministic constructions ba... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
we refer the reader to [BFMW13] for the details the construction consists of picking
rows of the p × p Discrete Fourier Transform (with p ∼= 1 mod 4 a prime) with indices corresponding
to quadratic residues modulo p. Just by coherence considerations this construction is known to be
RIP for s
, which would be predicted ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
known and there is no known (polynomial time) method
that is known to construct such partitions.
Open Problem 6.5 Give a (polynomial time) construction of the tight frame partition satisfying the
properties required in the Kadison-Singer problem (or the related Weaver’s conjecture).
30We note that the quadratic residue... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
parallel (at
the same time). This means that one has to non-adaptively design T tests (meaning subsets of the n
individuals) from which it is possible to detect the infected individuals, provided there are at most k
of them. Constructing these sets is the main problem in (Combinatorial) Group testing, introduced
by Rob... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
variable, A0, . . . , Ak. The probability that A0 is
contained in the union of A1 through Ak is given by
Pr [A0 ⊆ (A1 ∪ · · · ∪ Ak)] =
(cid:16)
1 − p(1 − p)k
(cid:17)
T
.
This is minimized for p = 1
k+1 . For this choice of p, we have
1 − p(1 − p)k = 1 −
1
k + 1
(cid:18)
1 −
(cid:19)
k
1
k + 1
Given that there are n su... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
17)
(cid:16)
− log
1 − 1
k+1 e−1 k+1
k
(cid:17) =
(cid:16)
(cid:1)(cid:17)
log
k(cid:0) n
k+1
− log (1 − (ek)−1)
=
O(k2 log(n/k)),
where the last inequality uses the fact that log
the Taylor expansion − log(1 − x−1)−1 = O(x)
(cid:16)(cid:0) n
k+1
(cid:1)(cid:17)
= O (cid:0)k log (cid:0) n
k
(cid:1)(cid:1)
due to Stirli... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
. For that, we will need a coup
family of sets. Let F denote the family of sets F in the definition of A1. More precisely,
le of auxiliary
F := {F ∈ [T ] : |F | = u and ∃!A ∈ A : F ⊆ A} .
By construction |A1| ≤ |F|
Also, let B be the family of subsets of [T ] of size u that contain an element of A0,
B = {B ⊆ [T ] : |B| ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:48)
= (cid:12)
(cid:12) .
(cid:12)
This implies that A
| 0| ≤ |B|
.
Because A satisfies the k-disjunct property, no two sets in A can contain eachother. This implies
that the families B and F of sets of size u are disjoint which implies that
|A0 ∪ A1| = |A0| + |A1| ≤ |B| + |F| ≤
(cid:19)
.
(cid:18)
T
u
Let A2 := A \ (A... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1 + u(k
−
j).
(cid:12)
(cid:12)
(cid:12)
(cid:12) k
(cid:91) (cid:12)
(cid:12)
Aj
(cid:12) =
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
j=0
(cid:88)
j=0
,...,k
(cid:12)
(cid:12)
(cid:12)Aj \
(cid:12)
(cid:12)
(cid:12)
(cid:91)
Ai
0 i<j
≤
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:88)
≥
j=0,...,k
[1 + ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
:18)
T
k2
(cid:19)
log k
,
which concludes the proof of the theorem.
(cid:50)
We essentially borrowed the proof of Theorem 7.2 from [Fur96]. We warn the reader however that
the notation in [Fur96] is drasticly different than ours, T corresponds to the number of people and n
to the number of tests.
There is another upper... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
, which implies that the union of that subfamily is the same as the
union of the same subfamily together with A. This means that a measurement system that is able to
uniquely determine a group of k infected individuals must be k − 1-disjunct.
7.2 Some Coding Theory and the proof of Theorem 7.3
In this section we (very)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1(cid:54)=c2∈C
ω(C) = min ∆(c).
c∈C\{0}
115 We say that a linear code C is a [N, m, d]q code (where N is the blocklenght, m the dimension, d
the distance, and Fq the alphabet.
One of the main goals of the theory of error-correcting codes is to understand the possible values
of rates, distance, and q for which codes exi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
GV bound (65).
1. Construct an explicit (deterministic) binary code (q = 2) satisfying the
2. Is the GV bound tight for binary codes (q = 2)?
7.2.1 Boolean Classification
A related problem is that of Boolean Classification [AABS15]. Let us restrict our attention to In
error-correcting codes one wants to build a linear co... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
1) goes to zero as n goes to infinity.
This is established in [AABS15] for β ≥ 2α but open in general.
7.2.2 The proof of Theorem 7.3
Reed-Solomon codes[RS60] are [n, m, n − m + 1]q codes, for m ≤ n ≤ q. They meet the Singleton
bound, the drawback is that they have very large q (q > n). We’ll use their existence to prov... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
Now we pick q ≈ 2k log n (q has to be a prime but there is always a prime between this number and
its double by Bertrand’s postulate (see [?] for a particularly nice proof)). Then m = log n (it can be
taken to be the ceiling of this quantity and then n gets updated accordingly by adding dummy sets).
log k
log q
32This ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
multiplications is that the addition operation is replaced by
binary “or”, meaning 1 ⊕ 1 = 1.
This means that we are essentially solving a linear system (with this non-standard multiplication).
Since the number of rows is T = O(k2 log(n/k)) and the number or columns n (cid:29) T the system is
underdetermined. Note that... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
and
5
(ij, kl) ∈ E G⊕2
(cid:0)
(cid:1)
if both a) i = k or i, k
∈ E and b) j
=
l or j, l
The above observation can then be written as α C⊕2
(cid:1)
= 5. This motivates the definition of
∈ E hold.
(cid:0)
5
Shannon Capacity [Sha56]
(cid:16)
G⊕k(cid:17) 1
k .
θS (G) sup
k
Lovasz, in a remarkable paper [Lov79], showed that... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
probability of 1 for each
Draw a random binary string
(cid:1)
bit (independently), define D n; 1 has the number of times the receiver needs to receive the message
2
so that she can decode the message exactly, with high probability.
(with independent corruptions)
It is easy to see that D n; 1 ≤ 2n, since roughly once in ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
with non-negative weights wij on
the edges, find a set S ⊂ V for which cut(S) is maximal. Goemans and Williamson [GW95] introduced
an approximation algorithm that runs in polynomial time and has a randomized component to it, and
is able to obtain a cut whose expected value is guaranteed to be no smaller than a particula... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/5f0f7205d1cf274e80d77345a7edbf2a_MIT18_S096F15_TenLec.pdf |
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