text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
right hand side to recover the topic model, provided that A has
full column rank. In fact, we can compute α0 from our samples (see [8]) but we will
focus instead on proving the above identity.
48
CHAPTER 3. TENSOR METHODS
Moments of the Dirichlet
The main identity that we would like to establish is jus... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
third ball is color k. The probably that the first ball is color i is α
i and since
we place it back with one more of its own color, the probability that the second ball
αk
is color i as well is
.
α0+2
It is easy to check the above formulas in the other cases too.
. And the probability that the third ball is color ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
, the i, i, i numerator of M ⊗ µ is
2(αi + 1). The case that requires some care is:
αi
Claim 3.5.9 If i = k, Ri,i,k = 0
(cid:54)=
The reason this case is tricky is because the terms M ⊗ µ(all three ways) do not
all count the same. If we think of µ along the third dimension of the tensor then
the ith topic occurs ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
polynomial time algorithm to learn a topic matrix
AA that is t close to the true A in a Latent Dirichlet Allocation model, provided we
are given at least poly(n, 1/t, 1/σr, 1/αmin) documents of length at least thee, where
n is the size of the vocabulary and σr is the smallest singular value of A and αmin
is the sma... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
phones and N conversations going on in an room.
Each microphone hears a superposition of the conversations given by the
corresponding rows of A. If we think of the conversations as independent
and memoryless, can we disentangle them?
Such problems are also often referred to as blind source separation. We will follo... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
approach was to consider higher order tensors. This time we will proceed
in a different manner.
Since M > 0 we can find B such that M = BBT . How are B and A related?
In fact, we can write
BBT = AAT ⇒ B−1AAT (B−1)T = I
and this implies that B−1A is orthogonal since a square matrix times its own trans
pose is the id... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
�v�2=1
What are its local minima?
�
�
E (v T x)4 = E
r
(vixi)4 + 6
r
(vixi)2(vj xj )2 =
r
v 4
i
=
i
E(x 4
i ) + 6
r
i
i v 2
v 2
j + 3
r
ij
v 4
i − 3
r
r
v 4
i + 3(
v 2
i )
ij
i
r
v 4
i
=
i
i
(cid:0)
i
− 3 + 3
(cid:1)
E x 4
i
52
CHAPTER 3. TENSOR METHODS
Hence the local ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the themselves the sum of the cumulants of Xi and Xj .
This is precisely the property we exploited here.
Chapter 4
Sparse Recovery
In this chapter we will study algorithms for sparse recovery: given a matrix A and
a vector b that is a sparse linear combination of its columns – i.e. Ax = b and x is
sparse – when ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
is to prove that finding the sparsest solution to a linear system
is hard. We will begin with the related problem:
Problem 1 (P) Find the sparsest non-zero vector x in a given subspace S
Khachiyan [81] proved that this problem is N P -hard, and this result has many
interesting applications that we will discuss later... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
determinant of the matrix whose columns are {Γw(αi)}i∈I .
Then the proof is based on the following observations:
(cid:0) (cid:1)
(a) The determinant is a polynomial in the variables αi with total degree n
2
+ 1,
which can be seen by writing the determinant in terms of its Laplace expansion
(see e.g. [74]).
(b) M... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Definition 4.1.3 A set of m vectors in Rn is in general position if every set of at
most n vectors is linearly independent.
From the above reduction we get that it is hard to decide whether a set of m
vectors in Rn is in general position or not (since there is an I with |I| = n whose
submatrix is singular if and onl... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
to Ax = 0. •
−i
−i
56
CHAPTER 4. SPARSE RECOVERY
4.2 Uniqueness and Uncertainty Principles
Incoherence
Here we will define the notion of an incoherent matrix A, and prove that if x is
sparse enough then it is the uniquely sparsest solution to Ax = b.
Definition 4.2.1 The columns of A ∈ Rn×m are µ-incoherent ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
where I ∈ Rn×n is the identity matrix and D ∈ Rn×n is the DFT matrix.
(i−1)(j−1)
In particular, Dij = w √
. This is often referred to as the
n
spikes-and-sines matrix. It is not hard to see that µ = √1
n
where w = e
here.
i 2π
n
Uncertainty Principles
The important point is that if A is incoherent, then if x i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
entry of V T U has absolute value at most
µ(A) and so |βT (V T U )α| ≤ µ(A)IαI1IβI1. Using Cauchy-Schwarz it follows that
IαI1 ≤ IαI0IαI2 and thus
(cid:112)
2
(cid:112)
IbI2
2 ≤ µ(A) IαI0IβI0IαI2IβI2
(cid:112)
1
Rearranging, we have µ(
A)
we get µ
2 ≤ IαI0 + IβI0 and this completes the proof. •
≤ IαI0IβI0. Fin... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
= IαyI0+IβyI0 ≥ µ
1 and so xA has strictly more non-zeros
It is easy to see that IxAI0 ≥ IyI0 − IxI0 > µ
than x does, and this completes the proof. •
Indeed, a similar statement is true even if A is an arbitrary incoherent matrix (in
stead of a union of two orthonormal bases). We will discuss this extension furthe... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
∈ ker(A). However IyI0 ≥ r + 1 because every set of r columns of A
is linearly independent, by assumption. Then IxAI0 ≥ IyI0 − IxI0 ≥ r/2 + 1 and so
xA has strictly more non-zeros than x does, and this completes the proof. •
In fact, if A is incoherent we can lower bound its Kruskal rank (and so the
proof in the pr... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
two.
1
Corollary 4.2.7 Suppose A is µ-incoherent. If Ax = b and IxI0 < 2
, then x is
µ
the uniquely sparsest solution.
There are a number of algorithms that recover x up to the uniqueness threshold in
the above corollary, and we will cover one such algorithm next.
4.3. PURSUIT ALGORITHMS
59
4.3 Pursuit Algori... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
algorithm selects is in T .
(b) Each index j gets chosen at most once.
These two properties immediately imply that orthogonal matching pursuit
recovers the true solution x, because the residual error r will be non-zero until S =
T , and moreover the linear system AT xT = b has a unique solution (since otherwise
x ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
columns of A
|yk ' +1| = 0, |yk ' +2| = 0, . . . , |ym| = 0.
where k ' ≤ k. Hence supp(y) = {1, 2, . . . , k ' } ⊆ T . Then to ensure that we pick
j ∈ T , a sufficient condition is that
(4.1)
�
−1
(cid:96)
|� A1, r
�
−1
(cid:96)
�| > |� Ai, r
�|
for all i ≥ k ' + 1.
We can lower-bound the left-hand side of (4.1... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
k '
r
�
=1
(cid:96)
y A , Ai ≤ |y1|
(cid:96)
�
(cid:96)
�
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
k '
r
|� A , Ai �| ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
uniqueness. However in the case where A = [U, V ]
and U and V are orthogonal, we proved a uniqueness result that is better by a factor
of two. There is no known algorithm that matches the best known uniqueness bound
there, although there are better algorithms than the one above (see e.g. [55]).
Matching Pursuit
We... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
(cid:18)
Fa,b =
exp
1
√
n
i2π(a − 1)(b − 1)
n
We can write ω = ei2π/n, and then the first row is √ [1, 1, . . . , 1]; the second row is
√1
1
n
[1, ω, ω2 , . . .], etc.
n
We will make use of following basic properties of F :
62
CHAPTER 4. SPARSE RECOVERY
(a) F is orthonormal: F H F = F F H , where ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
in the
Fourier representation:
Corollary 4.4.3 Let z = c ∗ x; then zA = Ac 8 xA, where 8 indicates coordinate-wise
multiplication.
Proof: We can write z = M cx = F H diag(Ac)F x = F H diag(Ac)xA = F H (Ac 8 xA), and
this completes the proof. •
We introduce the following helper polynomial, in order to describe Pro... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
, the zeros of vA correspond roots of p, and hence non-zeros of x. Conversely,
the non-zeros of vA correspond to zeros of x. We conclude that x 8 vA = 0, and so:
Corollary 4.4.7 M xv = 0
x
Proof: We can apply Claim 4.4.2 to rewrite x 8 vA = 0 as xA ∗ v = A0 = 0, and this
implies the corollary. •
Let us write out t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
first 2k values of the DFT of x. However consider the k × k + 1 submatrix
64
CHAPTER 4. SPARSE RECOVERY
whose top left value is xAk+1 and whose bottom right value is xAk. This matrix only
involves the values that we do know!
Consider
⎡
xk−1
A
xk
A
.
. .
⎢
⎣
x2k−1 x2k−1
A
A
. . .
x1
A
. . .
xk
A
⎡
⎤
⎢
⎥... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
system to find the non-zero values of x.
4.5 Compressed Sensing
that has an almost linear
Here we will give stable algorithms for recovering a signal
(in the number of rows of the sensing matrix) number of non-zeros. Recall that the
Kruskal rank of the columns of A is what determines how many non-zeros we can
allo... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Note that we will note require that w is k sparse. However if x is exactly
k sparse, then any w satisfying the above condition must be exactly equal to x
and hence this new recovery goal subsumes our exact recovery goals from previous
lectures (and is indeed much stronger).
The natural (but intractable) approach is... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
will make use of certain geometric properties of Γ (that hold
almost surely) in order to prove that basis pursuit works:
66
CHAPTER 4. SPARSE RECOVERY
Definition 4.5.6 Γ ⊆ Rn is an almost Euclidean subsection if for all v ∈ Γ,
1
√ IvI1 ≤ IvI2 ≤ √ IvI1
n
C
n
Note that the inequality √1
IvI1 ≤ IvI2 holds for all... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
S = n/C 2 .
Claim 4.5.7 Let v ∈ Γ, then either v = 0 or |supp(v)| ≥ S.
Proof:
IvI1 =
r
j∈supp(v)
|vj | ≤
(cid:112)
|supp(v)| · IvI2 ≤
(cid:112)
C
|supp(v)| √
n
IvI1
where the last inequality uses the property that Γ is almost Euclidean. The last
inequality implies the claim. •
4.5. COMPRESSED SENSI... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Λ denote the restriction of v to coordinates in
Λ. Similarly let vΛ denote the restriction of v to Λ.
¯
Claim 4.5.9 Suppose v ∈ Γ and Λ ⊆ [n] and |Λ| < S/16. Then
IvΛI1 <
IvI1
4
IvΛI1 ≤
(cid:112)
|Λ|IvΛI2 ≤
(cid:112)
C
|Λ|√
n
IvI1
Proof:
•
Hence not only do vectors in Γ have a linear number of non-zeros, b... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
This implies the
lemma. •
Hence we can use almost Euclidean subsections to get exact sparse recovery up to
(cid:16)
IxI0 = S/16 = Ω(n/C 2) = Ω
(cid:17)
n
log n/m
Next we will consider stable recovery. Our main theorem is:
Theorem 4.5.11 Let Γ = ker(A) be an almost Euclidean subspace with parameter
n
C. Let S =... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
≤ Ix − wI1 + 2σ S (x) .
16
1
2
This completes the proof. •
Notice that in the above argument, it was the geometric properties of Γ which
played the main role. There are a number of proofs that basis pursuit works, but the
advantage of the one we presented here is that it clarifies the connection between
the clas... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
weaker guarantees, such as ∀0 = v ∈ Γ, supp(v) =
Ω(n), but these do not suffice for compressed sensing since we also require that the
f1 weight is spread out too.
(cid:54)=
Chapter 5
Dictionary Learning
In this chapter we will study dictionary learning, where we are given many examples
b1, b2, ..., bp and we wou... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
has full column rank
(b) A is incoherent
71
72
(c) A is RIP
CHAPTER 5. DICTIONARY LEARNING
We will present an algorithm of Spielman, Wang and Wright [108] that succeeds
(under reasonable distributional assumptions) if A is full rank, and if each bi is a
n) columns in A. Next, we will give an algorithm
linear ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
and compression.
Dictionary learning is also used in the design of some deep learning architectures.
Popular approaches to solving this problem in practice are variants of the standard
alternating minimization approach. Suppose the pairs (xi, bi) are collected into the
columns of matrices X and B respectively. Then... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. Then an important rationale for preferring prov
able algorithms here is that if the examples are generated from an easy dictionary,
our algorithms will really learn an easy dictionary (but for the above heuristics this
need not be the case).
5.2 Full Rank Dictionaries
Here we present a recent algorithm of Spielm... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
not independent.
Outline
The basic idea is to consider the row space of B which we will denote by R = {wT B}.
Note that A−1B = X and hence the rows of X are contained in R. The crucial
observation is:
Observation 5.2.2 The rows of X are the sparsest non-zero vectors in R.
Of course finding the sparsest non-zero ve... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
We will prove that the solution to (Q1) is sparse, in particular supp(z) ⊆
supp(c).
• And for sparse z we will have that IzT XI1 ≈ IzI1 (after an appropriate
scaling). Hence we can instead analyze the linear program:
(Q1 ' )
min IzI1 s.t. c z = 1
T
Note that |supp(z)| = 1 if c has a coordinate that is strictly th... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
5.2.3 If R is a set of rows and C is a set of columns, let XC
submatrix of X that is the intersection of those rows and columns.
R be the
Let S be the set of columns of X that have a non-zero entry in J. That is
S = {j|X J = 00}. We now compute:
(cid:54)=
j
T
Iz∗ XI1 = Iz∗ XS I1 + Iz∗ XS I1
T
T
T
≥ Iz XS I1 − Iz... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
an ε-net of all possible z1’s and conclude that
T XI1 <
(5.1) holds for all z1’s. This in turn implies that if z1 is non-zero then Iz0
T XI1 but this is a contradiction since we assumed that z∗ is an optimal solution
Iz∗
to (Q1). We conclude that z1 is zero and so supp(z∗) ⊆ supp(c), as desired.
Step 2
We wish t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
5.3. OVERCOMPLETE DICTIONARIES
77
Step 3
We can now put everything together. Since c = xi, when we solve (P 1) we will get
the ith row of X up to scaling. If we solve (P 1) repeatedly, we will find all rows of
X (and can delete duplicates since now two rows of X will be scaled copies of each
other).
Finally we ca... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
above model. The running time and sample complexity are poly(n, m).
√
Recall that methods like K-SVD rely on the intuition that if we have the true
dictionary A, we can find X and if we have X we can find a good approximation
to A. However the trouble in analyzing these approaches is that they start from a
dictionar... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
it
should be reasonably likely that there is also an edge.
We can directly compute the above graph given our examples and the basic
idea is that we can hope to learn the support of X by finding the communities. The
key point is that this departs from standard clustering problems precisely because
each sample xi has... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
sub-Gaussian random variables, which satisfy E[Xi] = 0 and V ar[Xi] = 1.
5.3. OVERCOMPLETE DICTIONARIES
79
Let M be a symmetric n × n matrix. Then, for every t ≥ 0 we have:
(cid:18)
P[|x T M x − trace(M )| > t] ≤ 2 exp −c min
(cid:18)
Let Si and Sj be disjoint. Set N = (AT A)S
i
S
j
�
M =
0
1 N T
2
1
N ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
would need to solve
sparse recovery even if we knew A.
3
Community Finding
Consider the communities Cj = {bi|Si � j}. Then for each pair b1, b2 ∈ Cj there is
an edge (b1, b2) with probability at least 1
, and moreover our intersection graph can
2
be covered by m dense communities {Cj }j . We will introduce the b... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
if i ∈ S1 ∩ S2 ∩ S3, then the probability that S4
contains i is k/m. Next we prove the upper bound. Let a = |S1 ∩ S2|, b = |S1 ∩ S3|
and c = |S2 ∩ S3|. Then:
Lemma 5.3.5 If S1 ∩S2 ∩S3 = ∅ the probability that (b4, bi) is an edge for i = 1, 2, 3
is at most
(cid:18)
O
k6
3
m
+
k3(a + b + c)
2
m
(cid:19)
Proof: We ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
is easy to see that with high probability for any
i, j, |Si ∩ Sj | = O(1) and hence we want (roughly)
k
m
>>
k6
3
m
+
k3
2
m
which is true if k < m2/5 . When this holds, for every triple b1, b2, b3 we will be able
to determine whether or not S1 ∩ S2 ∩ S3 is empty with high probability by counting
how many othe... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
is not an identifying pair but rather S1 ∩ S2 has more than
one element we would have instead C1,2 = ∪j∈S1∩S2 Cj in which case the set C1,2 will
be deleted in the last step. This algorithm outputs the correct communities with
high probability if k ≤ c min( n/µ log n, m2/5). In [15] the authors give a higher
order a... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
a mixture of k Gaussians, we would like
to give an efficient algorithm to learn its parameters using few samples. If these
parameters are accurate, we can then cluster the samples and our error will be
nearly as accurate as the Bayes optimal classifier.
6.1 History
The problem of learning the parameters of a mixture ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Gaussians. Recall
that for a univariate Gaussian we have that its density function is given by:
N (µ, σ2) = √
1
2πσ2
exp
−(x − µ)2
2σ2
The density of a multidimensional Gaussian in Rn is given by:
N (µ, Σ) =
1
(2π)n/2det(Σ)1/2
exp
−(x − µ)�Σ−1(x − µ)
2
Here Σ is the covariance matrix. If Σ = In and µ = 00 the... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
1
polynomial (Pr) in the unknown parameters.
6.1. HISTORY
85
Image courtesy of Peter D. M. Macdonald. Used with permission.
Figure 6.1: A fit of a mixture of two univariate Gaussians to the Pearson’s data on
Naples crabs, created by Peter Macdonald using R
Pearson’s Sixth Moment Test: We can estimate Ex←F... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
0.580.600.620.640.660.680.7005101520
86
CHAPTER 6. GAUSSIAN MIXTURE MODELS
Expectation-Maximization
Much of modern statistics instead focuses on the maximum likelihood estimator,
which would choose to set the parameters to as to maximize the probability that
the mixture would generate the observed samples. Unf... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
no “overlap”. The next generation algorithms are based on algebraic
ideas, and avoid clustering altogether.
Before we proceed, we will discuss some of the counter-intuitive properties of
high-dimensional Gaussians. To simplify the discussion, we will focus on spherical
Gaussians N (µ, σ2I) in Rn .
Fact 6.2.1 The m... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
j with high probability, but will contract distances between
samples from the same component and make each component closer to spherical,
thus making it easier to cluster. We can then cluster all of the samples into which
component generated them, and then for each cluster we can choose the empirical
mean and empir... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
In fact, any set of vectors in which all
but one is uniformly random from a sphere are nearly orthogonal.
Now we can compute:
Ix − x ' I2 ≈ Ix − µ1I2 + Iµ1 − x
≈ 2nσ2 ± 2σ2 n log n
(cid:112)
' I2
And similarly:
Ix − yI2 ≈ Ix − µ1I2 + Iµ1 − µ2I2 + Iµ2 − yI2
≈ 2nσ2 + Iµ1 − µ2I2 ± 2σ2 n log n
(cid:112)
A
Hence if ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
σ2I. Let
x ∼ F be a random sample from the mixture, then we can write x = c + z where
z ∼ N (0, σ2In) and c is a random vector that takes the value µi with probability wi
for each i ∈ [k]. So:
E[xx T ] = E[cc T ] + E[zz T ] =
k
r
wiµiµi + σ2In
�
(cid:62)
Hence the top left singular vectors of E[xxT ] whose singu... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
[45], [53], [18], [117], [1], [32] have focused
on clustering; can we give efficient learning algorithms even when clustering is im
possible? Consider a mixture of two Gaussians F = w1F1 + w2F2. The separation
conditions we have considered so far each imply that dT V (F1, F2) = 1 − o(1). In
particular, the components... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Choose (x, y) from the best coupling between F1 and F2
(b) If x = y, output x
90
CHAPTER 6. GAUSSIAN MIXTURE MODELS
(c) Else output x with probability w1, and otherwise output y
This procedure generates a random sample from F , but for half of the samples we
did not need to decide which component generated it ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
is smooth, we would need to take an exponential number of samples in
order to guarantee that FA = G ∗ N (0, σ2I) is close to F .
(b) Proper Density Estimation
Here, our goal is to find a distribution FA ∈ C where dT V (F, FA) ≤ ε. Note that if
C is the set of mixtures of two Gaussians, then a kernel density estimate... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
be close as mixtures too: dT V (F, F ) ≤ 4ε.
However we can have mixtures F and F that are both mixtures of k Gaussians,
A
are close as distributions
basis. It
is better to learn F on a component-by-component basis than to do only proper
density estimation, if we can. Note that if FA is ε-close to F , then even wh... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
.4 Clustering-Free Algorithms
Recall, our goal is to learn FA that is ε-close to F . In fact, the same definition can
be generalized to mixtures of k Gaussians:
6.4.1
Definition
component-by-component basis) to F if there is
{1, 2, ..., k} so that for all i ∈ {1, 2, ..., k}:
We will say that a mixture F =
(cid:80... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
learn wi or wj because Fi and Fj entirely
overlap.
Again, we need a quantitive lower bound on dT V (Fi, Fj ), say dT V (Fi, Fj ) ≥ ε for
each i = j so that if we take a reasonable number of samples we will get at least
one sample from the non-overlap region between various pairs of components.
(cid:54)=
Theorem 6.... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the parameters be additively close
which is the goal in [23]. The benefit is that the algorithm in [23] works for more
general learning problems in the one-dimensional setting, and we will describe this
algorithm in detail at the end of this chapter.
Throughout this section, we will focus on the k = 2 case since thi... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
set y = B−1(x − µ), and it is easy to see that E[y] = 0
and E[yyT ] = B−1M (B−1)T = I. •
Our goal is to learn an additive ε approximation to F , and we will assume that F
has been pre-processed so that it is in isotropic position.
Outline
We can now describe the basic outline of the algorithm, although there will ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
some notation:
Definition 6.4.6 dp(N (µ1, σ2), N (µ2, σ2)) = |µ1 − µ2| + |σ2 − σ2
2|
1
1
2
We will refer to this as the parameter distance. Ultimately, we will give a univariate
algorithm for learning mixtures of Gaussians and we would like to run it on projr[F ].
Problem 4 But what if dp(projr[F1], projr[F2]) is e... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
position and wi ≥ ε and dT V (F1, F2) ≥ ε, then
with high probability for a random r
dp(proj [F1], proj [F2]) ≥ 2ε3 = poly(1/n, ε)
r
r
6.4. CLUSTERING-FREE ALGORITHMS
95
Note that this lemma is note true when F is not in isotropic position (e.g. consider
the parallel pancakes example), and moreover when generaliz... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
to solve for the
parameters of F1, but when we project F onto different directions (say, r and s)
we need to pair up the components from these two directions. The key observation
is that as we vary r to s the parameters of the mixture vary continuously. See
Figure ??. Hence when we project onto r, we know from the i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
we encounter the final problem in the high-dimensional case: Suppose we choose
r randomly and for s1, s2, ...., sp we learn the parameters of the projection of F onto
these directions and pair up the components correctly. We can only hope to learn the
parameters on these projection up to some additive accuracy ε1 (an... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
polynomial number of mixtures to learn an ε-close estimate FA!
But we still need to design a univariate algorithm, and next we return to
Pearson’s original problem!
6.5 A Univariate Algorithm
Here we will give a univariate algorithm to learning the parameters of a mixture of
two Gaussians up to additive accuracy ε... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the first six moments of F (Θ) from
enough random examples, and output AΘ if its first six moments are each within an
additive τ of the observed moments. (This is a slight variant on Pearson’s sixth
moment test).
It is easy to see that if we take enough samples and set τ appropriately, then
if we round the true para... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
2τ
Mr( A Mr Mr
(cid:12)
(cid:124)
(cid:12)
(cid:12)
(cid:12)
(cid:125)
(cid:12)
(cid:12)
(cid:12)
(cid:125)
(cid:12)
(cid:12)
(cid:12)
(cid:124)
(cid:12)
(cid:12)
(cid:12)
(cid:123)(cid:122)
≤τ
(cid:123)(cid:122)
≤τ
98
CHAPTER 6. GAUSSIAN MIXTURE MODELS
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
In fact, let us consider the following thought experiment. Let f (x) = F (x) −
F (x) be the point-wise difference between the density functions F and F . Then,
A
the heart of the problem is: Can we prove that f (x) crosses the x-axis at most six
times? See Figure 6.2.
A
Lemma 6.5.3 If f (x) crosses the x-axis at mos... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
to prove is that F (x) − FA(x) has at most six zero crossings. Let
us prove a stronger lemma by induction:
2, x) be a linear combination of k Gaus
Lemma 6.5.4 Let f (x) =
sians (αi can be negative). Then if f (x) is not identically zero, f (x) has at most
2k − 2 zero crossings.
k
=1 αiN (µi, σi
i
(cid:80)
We wi... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
first six moments of a mixture of two Gaussians exactly, then we would know its
parameters exactly too. Let us prove the above lemma by induction, and assume
that for any linear combination of k = 3 Gaussians, the number of zero crossings is
100
CHAPTER 6. GAUSSIAN MIXTURE ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
this last step does
not increase the number of zero crossings! See Figure 6.3(d).
This proves that
(cid:110)
Mr( A
Θ) = Mr
(cid:111)
(Θ)
,
r = 1, 2, ..., 6
has only two solutions (the true parameters and we can also interchange which is
component is which). In fact, this system of polynomial equations is also sta... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
X defined as
!
n
converges in a neighborhood of zero, it uniquely determines the probability distribu
tion, i.e.
(cid:80)
∀r, Mr(Θ) = Mr Θ) ⇒ F (Θ) = F (Θ)A .
( A =
Our goal is to show that for any polynomial family, a finite number of its moments
suffice. First we introduce the relevant definitions:
Definition 6.6.3 G... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
tion converges in a neighborhood of zero, there exists N such that
F (Θ) = F (Θ)A if and only if Mr(Θ) = Mr Θ) ∀r ∈ 1, 2, · · · , N
( A
Proof: Let Qr(Θ, Θ) = Mr(Θ) − Mr Θ). Let I1 = Q1 , I2 = Q1, Q2 , · · · .
( A
This is our ascending chain of ideals in R[Θ, Θ]. We can invoke Hilbert’s basis
A
(cid:104)
�
(cid:105)... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
contradiction, but
more fundamentally it is not possible to give a bound on N that depends only on
the choice of the ring. Consider the following example
Example 1 Consider the Noetherian ring R[x]. Let Ii = xN −i
for i = 0, · · · , N .
It is a strictly ascending chain of ideals for i = 0, · · · , N . Therefore, ev... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
:
Definition 6.6.7 A set S is semi-algebraic if there exist multivariate polynomials
p1, ..., pn such that
S = {x1, ..., xr|pi(x1, ..., xr) ≥ 0}
or if S is a finite union or intersection of such sets.
104
CHAPTER 6. GAUSSIAN MIXTURE MODELS
Theorem 6.6.8 (Tarski) The projection of a semi-algebraic set is semi-... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
a
fixed δ we can choose ε to be strictly greater than zero and moreover the polynomial
relationship between ε and δ only holds if δ is sufficiently small. However these
technical issues can be resolved without much more work, see [23] and the main
result is the following. •
Corollary 6.6.10 If |Mr(Θ) − Mr Θ)| ≤
( A
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
some of the entries of M to fill in the rest of M .
Let us be more precise: There is an unknown matrix M ∈ Rn×m whose rows
represent users and whose columns represent movies in the example above. For each
(i, j) ∈ Ω ⊆ [n] × [m] we are given the value Mi,j . Our goal is to recover M exactly.
Ideally, we would like to... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
it would
be more believable if the probability we observe Mi,j depends on the value itself.
Alternatively, a user should be more likely to rate a movie if he actually liked it.
In order to understand the second assumption, suppose Ω is indeed uniformly
random. Consider
�
(cid:21)
�
(cid:20)
M = Π
Ir
0
where Π i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
1.2 Suppose Ω is chosen uniformly at random. Then there is a poly
nomial time algorithm to recover M exactly that succeeds with high probability if
m ≥ max(µ1
2 , µ0)r(n + m) log2(n + m)
7.2. NUCLEAR NORM
107
The algorithm in the theorem above is based on a convex relaxation for the rank
of a matrix called the n... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the basis for our algorithms for
matrix completion. We will follow a parallel outline to that of compressed sensing.
In particular, a natural starting point is the optimization problem:
(P 0)
min rank(X) s.t. Xi,j = Mi,j for all (i, j) ∈ Ω
This optimization problem is N P -hard. If σ(X) is the vector of singular v... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
X, B =
product.
�
(cid:104)
(cid:105) (cid:80)
�
i,j Xi,j Bi,j = trace(X T B) denote the matrix inner-
Lemma 7.2.3 IXI∗ = max B �≤1 X, B .
�
(cid:105)
�
(cid:104)
�
(cid:107)
To get a feel for this, consider the special case where we restrict X and B to be
diagonal. Moreover let X = diag(x) and B = diag(b). Then I... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
, B , and the other direction is not much more difficult
(see e.g. [74]). •
�
(cid:105)
�
(cid:104)
�
(cid:107)
How can we show that the solution to (P 1) is M ? Our basic approach will
be a proof by contradiction. Suppose not, then the solution is M + Z for some Z
that is supported in Ω. Our goal will be to construc... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. ur+1, . . . , un is an arbitrary orthonormal basis of U ⊥ . Similarly
choose vr+1, . . . , vn so that v1, . . . , vn form an orthonormal basis for all of Rn . We
will be interested in the following linear spaces over matrices:
Definition 7.2.4 T = span{uivT | 1 ≤ i ≤ r or 1 ≤ j ≤ r or both}.
j
Then T ⊥ = span{uivT... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
the helper matrix Y and prove that if
such a Y exists, then M is the solution to (P 1). We require that Y is supported in
Ω and
(a) IPT (Y ) − U V T IF ≤
(cid:112)
r/8n
110
CHAPTER 7. MATRIX COMPLETION
(b) IPT ⊥ (Y )I ≤ 1/2.
We want to prove that for any Z supported in Ω, IM + ZI∗ > IM I∗. Recall,
we want t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
:107)
where in the last line we used the fact that M is orthogonal to U⊥V⊥
the fact that Y and Z have disjoint supports we can conclude:
T . Now using
IM + ZI∗ ≥ IM I∗ + Z, U V T + U⊥V⊥
(cid:104)
�
T − Y
(cid:105)
�
Therefore in order to prove the main result in this section it suffices to prove that
Z, U V T + U⊥V... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
[Z] and hence U⊥V⊥
T , PT ⊥ [Z] = IPT ⊥ [Z]I∗.
(cid:105)
�
(cid:104)
�
Now we can invoke the properties of Y that we have assumed in this section,
to prove a lower bound on the right hand side. By property (a) of Y , we have that
IPT (Y ) − U V T IF ≤
. Therefore, we know that the first term PT (Z), U V T −
r
�
PT... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
IPT ⊥ (Z)I∗ >
IPT (Z)IF to
n
2
complete the proof we started in the previous section. We will make use of an
approach introduced by Gross [67] and we will follow the proof of Recht in [103]
where the strategy is to construct Y iteratively. In each phase, we will invoke
concentration results for matrix valued rand... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
following toy
problem. Let uk be a random unit vector in Rd and let Xk = ukuk
T . Then it is easy
to see that ρ2 = 1/d. How many trials do we need so that
is close to the
identity (after scaling)? We should expect to need Θ(d log d) trials; this is even true
if uk is drawn uniformly at random from the standard bas... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
framework of the matrix Bernstein
inequality, but for a full proof see [103]. Note that E[PT RΩPT ] = PT
E[RΩ]PT =
m
RΩPT does not deviate too far from its
2 PT and so we just need to show that PT
n
expectation. Let e1, e2, . . . , ed be the standard basis vectors. Then we can expand:
PT (Z) =
r
�(cid:104)P
T (... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
7.3. QUANTUM GOLFING
113
Lemma 7.3.5 If Ω is chosen uniformly at random and m ≥ nr log n then with high
probability for any Z supported in Ω we have
IPT ⊥ (Z)I∗ >
(cid:114)
r
2n
IPT (Z)IF
Proof: Using Lemma 7.3.3 and the definition of the operator norm (see the remark)
we have
m
2
2n
Furthermore we can upper b... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
24n
2
IPT (Z)IF
All that remains is to prove that the helper matrix Y that we made use of
actually does exists (with high probability). Recall that we require that Y is sup
ported in Ω and IPT (Y ) − U V T IF ≤
r/8n and IPT ⊥ (Y )I ≤ 1/2. The basic idea
is to break up Ω into disjoint sets Ω1, Ω2, . . . Ωp, where ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
It is easy to see that Y =
of the hole
all i. Hence we can compute
PT (Yi) − U V T IF =
I
≤
�
(cid:13)
(cid:13)�
(cid:13)
Wi 1 − P
(cid:13)(cid:13)�
−
�
(cid:13)
n2
m
�(cid:13)
(cid:13)�(cid:13)
PT RΩPT
�
(cid:13)
(cid:13)
�
(cid:13)(cid:13)
(cid:13)
�
T RΩi Wi−1
(cid:13) =
�(cid:13)
(cid:13)
�
F
n2
m
�
(cid:13)... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Y )I ≤
IP T ⊥(Yi)
I
i
comes from the first term. For the full details see [103]. This completes the proof
that computing the solution to the convex program indeed finds M exactly, provided
that M is incoherent and |Ω| ≥ max(µ2
1, µ0)r(n + m) log2(n + m).
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[1] D. Achlioptas and F. McSherry. O... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
.
[7] Noga Alon. Tools from Higher Algebra. In Handbook of Combinatorics, pages
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[8] A. Anandkumar, D. Foster, D. Hsu, S. Kakade, Y. Liu. A spectral algorithm
for latent dirichlet allocation. In NIPS, pages 926–934, 2012.
[9] A. Anandkumar, R. Ge, D. Hsu and S. Kakade. A tensor spectral approach
t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
– provably In STOC, pages 145–162, 2012.
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[16] S. Arora, R. Ge, A. Moitra and S. Sa... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
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[24] Q. Berthet and P. Rigollet. Complexity theoretic lower bounds for sparse prin
cipal comp... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
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