text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
≤ W Θ) − W + W − Mr(Θ) ≤ 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. ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
different mixtures of two Gaussians, why is one of their
first six moments necessarily different? Our main goal is to prove this statement,
using the heat equation.
A
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 .... | 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)
r=1
6
r
(cid:12)
(cid:12)
|pr| Mr(Θ) − Mr( AΘ)
|
(cid:12)
(cid:12)
|
≤
And if the first six moments of F and FA match exactly, the right hand side is zero
which is a contradiction. •
So all we need to prove is that F (x) − FA(x) has at most six zero crossings. Let
us prove... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
intuitions for diffusion have consequences for convolution
– convolving a function by a Gaussian has the effect of smoothing it, and it cannot
create a new local maxima (and relatedly it cannot create new zero crossings).
Finally we recall the elementary fact:
Fact 6.5.6 N (0, σ1
2) ∗ N (0, σ2
2) = N (0, σ1
2 + σ2
2)... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
ians and by induction we know that it has at most four zero crossings. But
how many zero crossings can we add when we add back in the delta function? We
can add at most two, one on the way up and one on the way down (here we are
ignoring some real analysis complications of working with delta functions for ease of
p... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
Mr(Θ) is a polynomial in Θ = (θ1, θ2, ...., θk).
102
CHAPTER 6. GAUSSIAN MIXTURE MODELS
This definition captures a broad class of distributions such as mixtures models whose
components are uniform, exponential, Poisson, Gaussian or gamma functions. We
will need another (tame) condition on the distribution whic... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
I3 ⊆ · · · ,
Theorem 6.6.5 (Hilbert’s Basis Theorem) If R is a Noetherian ring, then R[X]
is also a Noetherian ring.
It is easy to see that R is a Noetherian ring, and hence we know that R[x] is also
Noetherian. Now we can prove that for any polynomial family, a finite number of
moments suffice to uniquely identify a... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
) =
A
N
r
pij (Θ, Θ)Qi(Θ, Θ)
A
A
i=1
for some polynomial pij ∈ R[Θ, Θ]. Thus, if Mr
then Mr(Θ) = Mr Θ) for all r and from Fact 6.6.2 we conclude that F (Θ) = F ( A
(Θ) = Mr Θ) for all r ∈ 1, 2, · , N ,
Θ).
( A
( A
A
·
·
The other side of the theorem is obvious. •
The theorem above does not give any finite bound o... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. Nevertheless, these tools apply much more broadly than the specialized
ones based on the heat equation that we used to prove that 4k − 2 moments suffice
for mixtures of k Gaussians in the previous section.
Systems of Polynomial Inequalities
In general, we do not have exact access to the moments of a distribution bu... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
6.6.9 There are fixed constants C1, C2, s such that if δ < 1/C1 then
ε(δ) < C2δ1/s.
Proof: It is easy to see that we can define H(ε, δ) as the projection of a semi-
algebraic set, and hence using Tarski’s theorem we conclude that H(ε, δ) is also
semi-algebraic. The crucial observation is that because H(ε, δ) is semi-a... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
ε);
we can take enough samples to estimate the first N moments within εs and search
over a grid of the parameters, and any set of parameters that matches each of the
moments is necessarily close in parameter distance to the true parameters.
Chapter 7
Matrix Completion
Here we will give algorithms for the ma... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
are uncorrelated with the standard basis (such a
matrix is called incoherent and we define this later)
In fact, we will see that there are efficient algorithms for recovering M exactly if
m ≈ mr log m where m ≥ n and rank(M ) ≤ r. This is similar to compressed
105
106
CHAPTER 7. MATRIX COMPLETION
sensing, where we... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
standard basis, we could avoid the above problem.
0
0
ΠT
Definition 7.1.1 The coherence µ of a subspace U ⊆ Rn of dimension dim(u) = r
is
n
r
max IPU eiI2 ,
i
where PU denotes the orthogonal projection onto U , and ei is the standard basis
element.
It is easy to see that if we choose U uniformly at random, t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
it as an analogue to the
f1 minimization approach that we used in compressed sensing. This approach was
first introduced in Fazel’s thesis [58], and Recht, Fazel and Parrilo [104] proved that
this approach exactly recovers M in the setting of matrix sensing, which is related
to the problem we consider here.
In a la... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
finding the sparsest solution
to a system of linear equations is also N P -hard, but we instead considered the
f1 relaxation and proved that under various conditions this optimization problem
recovers the sparsest solution. Similarly it is natural to consider the f1-norm of σ(X)
which is called the nuclear norm:
De... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
and B to be
diagonal. Moreover let X = diag(x) and B = diag(b). Then IXI∗ = IxI1 and
the constraint IBI ≤ 1 (the spectral norm of B is at most one) is equivalent to
IbI∞ ≤ 1. So we can recover a more familiar characterization of vector norms in
the special case of diagonal matrices:
IxI1 = max
� �
b ∞≤1
(cid:107... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
contradiction. Suppose not, then the solution is M + Z for some Z
that is supported in Ω. Our goal will be to construct a matrix B of spectral norm
at most one for which
IM + ZI∗ ≥ M + Z, B > IM I∗
�
(cid:105)
�
(cid:104)
Hence M + Z would not be the optimal solution to (P 1). This strategy is similar to
the one i... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
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 s.t. r + 1 ≤ i, j ≤ n}.. We have dim(T ) = r2 + 2(n − r)r and
dim(T ⊥) = (n − r)2 . Moreover we can define the linear operators that project into ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
) 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 to find a matrix B of spectral norm at most one so that M +Z, B > IM I∗.
Let U⊥ and V⊥ be singular vectors of PT ⊥ [Z]. Then consider ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
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 T − Y > 0. We can expan... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
of
PT ⊥ [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 ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
helper matrix Y and prove that with high probability
r
over Ω, for any matrix Z supported in Ω that 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... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
:80)
If d = 1 this is the standard Bernstein inequality. If d > 1 and the matrices Xk are
diagonal then this inequality can be obtained from the union bound and the standard
Bernstein inequality again. However to build intuition, consider the following toy
problem. Let uk be a random unit vector in Rd and let Xk = u... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
7.3.4 Here we are inter
operator on matrices. Let T be such an operator, then IT I is defined as
ested in bounding the operator norm of a linear
max IT (Z)
�(cid:107)Z (cid:107)�F 1
≤
IF
We will explain how this bound fits into the framework of the matrix Bernstein
inequality, but for a full proof see [103]. Note th... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
think of PT RΩPT as the sum of random operators of the form τa,b : Z →
W
(cid:11)
(cid:10)
T P
Z, P
T (eaeb ), and the lemma follows by applying the matrix Bernstein
T (eaeb )
the random operator (cid:80)
inequality to
.
T
τ
∈Ω a,b
(a,b)
We can now complete the deferred proof of part (a):
... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
)I2 ≥ IPT (Z)I2 −
2
n
F
F
m
2
2n
IZI2
F
We can use the fact that IZI2 = IPT ⊥ (Z)IF
m
4n IPT (Z)I2 We can now complete the proof of the lemma
2
F .
F
F
2 +IPT (Z)I2 and conclude IPT ⊥ (Z)I2 ≥
F
IPT ⊥ (Z)I∗
2 ≥ IPT ⊥ (Z)IF
2 ≥
•
>
r
2n
IPT (Z)I2
F
m
24n
2
IPT (Z)IF
All that remains is to prove that t... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
n
m
itively this means that at each step Yi+1 − Yi is an unbiased estimator for Wi and so
we should expect the remainder to decrease quickly (here we will rely on the concen
tration bounds we derived from the non-commutative Bernstein inequality). Now
114
CHAPTER 7. MATRIX COMPLETION
the direction
we ca... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
)
�
F
−
�
(cid:13)
(cid:13)
m
(cid:13)�(cid:13)
PT
(cid:13)�
2
n
≤
1
I
2
I
Wi−1 F
last
the
where
of
condition (a).
the remainder
inequality
decreases
follows from Lemma
geometrically and
7.3.3. Therefore the Frobenius norm
it is easy to guarantee that Y satisfies
The more technically involved part is sh... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
overcomplete dictionaries via alternating minimization
arxiv:1310.7991, 2013
[3] A. Agarwal, A. Anandkumar, P. Netrapalli Exact recovery of sparsely used
overcomplete dictionaries arxiv:1309.1952, 2013
[4] M. Aharon. Overcomplete Dictionaries for Sparse Representation of Signals.
PhD Thesis, 2006.
[5] M. Aharon, ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
mixed membership community models. In COLT, pages 867–881,
2013.
[10] A. Anandkumar, D. Hsu and S. Kakade. A method of moments for hidden
markov models and multi-view mixture models. In COLT, pages 33.1–33.34,
2012.
115
116
BIBLIOGRAPHY
[11] J. Anderson, M. Belkin, N. Goyal, L Rademacher and J. Voss. The more ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
incoherent and
overcomplete dictionaries. arxiv:1308.6273, 2013
[16] S. Arora, R. Ge, A. Moitra and S. Sachdeva. Provable ICA with unknown
gaussian noise, and implications for gaussian mixtures and autoencoders. In
NIPS, pages 2384–2392, 2012.
[17] S. Arora, R. Ge, S. Sachdeva and G. Schoenebeck. Finding overlappi... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
407–419, 2010.
[23] M. Belkin and K. Sinha. Polynomial learning of distribution families. In FOCS,
pages 103–112, 2010.
[24] Q. Berthet and P. Rigollet. Complexity theoretic lower bounds for sparse prin
cipal component detection. In COLT, pages 1046–1066, 2013.
BIBLIOGRAPHY
117
[25] A. Bhaskara, M. Charikar and... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
[31] A. Blum, A. Kalai and H. Wasserman. Noise-tolerant learning, the parity
problem, and the statistical query model. Journal of the ACM 50: 506-519,
2003.
[32] S. C. Brubaker and S. Vempala. Isotropic PCA and affine-invariant clustering.
In FOCS, pages 551–560, 2008.
[33] E. Candes and B. Recht. Exact matrix compl... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
and S. Rao. Beyond Gaussians: Spectral methods for learning
mixtures of heavy-tailed distributions. In COLT, pages 21–32, 2008.
118
BIBLIOGRAPHY
[39] S. Chen, D. Donoho and M. Saunders. Atomic decomposition by basis pursuit.
SIAM J. on Scientific Computing, 20(1):33–61, 1998.
[40] A. Cohen, W. Dahmen and R. DeVor... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
L. J. Schulman. A two-round variant of EM for gaussian
mixtures. In UAI, pages 152–159, 2000.
[47] G. Davis, S. Mallat and M. Avellaneda. Greedy adaptive approximations. J.
of Constructive Approximation, 13:57–98, 1997.
[48] L. De Lathauwer, J Castaing and J. Cardoso. Fourth-order Cumulant-based
Blind Identificatio... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
–2862, 1999.
[53] D. Donoho and P. Stark. Uncertainty principles and signal recovery. SIAM J.
on Appl. Math., 49(3):906–931, 1989.
BIBLIOGRAPHY
119
[54] D. Donoho and V. Stodden. When does nonnegative matrix factorization give
the correct decomposition into parts? In NIPS, 2003.
[55] M. Elad. Sparse and Redunda... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
. A. Servedio, and R. O’Donnell. PAC learning axis-aligned
mixtures of gaussians with no separation assumption. In COLT, pages 20–34,
2006.
[62] A. Frieze, M. Jerrum, R. Kannan. Learning linear transformations. In FOCS,
pages 359–368, 1996.
[63] A. Garnaev and E. Gluskin. The widths of a Euclidean ball. Sovieth Ma... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
uruswami, J. Lee, and A. Razborov. Almost euclidean subspaces of fn
1 via
expander codes. Combinatorica, 30(1):47–68, 2010.
120
BIBLIOGRAPHY
[70] R. Harshman. Foundations of the PARFAC procedure: model and conditions
for an ‘explanatory’ multi-mode factor analysis. UCLA Working Papers in
Phonetics, pages 1–84, ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
11, 1984.
[78] R. Impagliazzo and R. Paturi. On the complexity of k-SAT. J. Computer and
System Sciences 62(2):pp. 367–375, 2001.
[79] A. T. Kalai, A. Moitra, and G. Valiant. Efficiently learning mixtures of two
gaussians. In STOC, pages 553-562, 2010.
[80] B. Kashin and V. Temlyakov. A remark on compressed sensing.... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
objects by non-negative matrix
factorization. Nature, pages 788-791, 1999.
[86] D. Lee and H. Seung. Algorithms for non-negative matrix factorization. In
NIPS, pages 556–562, 2000.
[87] S. Leurgans, R. Ross and R. Abel. A decomposition for three-way arrays. SIAM
Journal on Matrix Analysis and Applications, 14(4):1... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
rank. In
SODA, pages 1454–1464, 2003.
[95] A. Moitra and G. Valiant. Setting the polynomial learnability of mixtures of
gaussians. In FOCS, pages 93–102, 2010.
[96] E. Mossel and S. Roch. Learning nonsingular phylogenies and hidden markov
models. In STOC, pages 366–375, 2005.
[97] B. Olshausen and B. Field. Spars... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
of arbitrary dis
tributions over large discrete domains. . In ITCS 2014.
[102] R. Raz. Tensor-rank and lower bounds for arithmetic formulas. In STOC,
pages 659–666, 2010.
[103] B. Recht. A simpler approach to matrix completion. Journal of Machine
Learning Research, pages 3413–3430, 2011.
[104] B. Recht, M. Fazel ... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
it generates under a
Markov model. Appl. Math. Lett. 7:19-24, 1994.
[110] A. Tarski. A decision method for elementary algebra and geometry. University
of California Press, 1951.
[111] H. Teicher. Identifiability of mixtures. Annals of Mathematical Statistics,
pages 244–248, 1961.
[112] J. Tropp. Greed is good: Alg... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
,odels.
Journal of Computer and System Sciences, pages 841–860, 2004.
[118] M. Wainwright and M. Jordan. Graphical Models, Exponential Families, and
Variational Inference. Foundations and Trends in Machine Learning, pages
1–305, 2008.
[119] P. Wedin. Perturbation bounds in connection with singular value decomposi... | https://ocw.mit.edu/courses/18-409-algorithmic-aspects-of-machine-learning-spring-2015/2af5365a3f0d24cc2ee9f787bbab14e9_MIT18_409S15_bookex.pdf |
18.725 Algebraic Geometry I
Lecture
4
Lecture 4: Grassmannians, Finite and Affine Morphisms
Remarks on last time
1. Last time, we proved the Noether normalization lemma: If A is a finitely generated k-algebra, then, A
contains B ∼= k[x1, . . . , xn] (free subring) such that A is a finitely generated B-module.
Question: Whe... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
b(cid:48)2) ∼= A1 = U2
Y
Y
(a) X = 0: a = , b = , (a = b2)
X
X
Y
∼= A1 = U1
Note that U1 ∩ U2
∼= A1 \ {0}.
By changing coordinates, we can take the degree 2 curve in P2 to be X 2 + Y 2 = Z 2. Connect points in
a quadric to a fixed point. In practice, we can work with the point (1 : 0 : 1). We identify the set of all
lin... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
icts the assumption that X is Noetherian.
Now we begin the proof of the proposition.
1
(cid:54)
(cid:54)
18.725 Algebraic Geometry I
Lecture
(cid:23)
Proof. Write X =
n
(cid:91)
i=1
Xi, where the Xi are closed irreducible subsets of X. Without loss of generality, we
can assume that none of the Xi are a subset of anoth... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
pec A) is the set of k-valued
functions on Spec A. Suppose that Spec A is irreducible. If f g = 0, then Zf ∪ Zg = Spec A, where Zf are
the zeros of f and Zg are the zeros of g. If Zf , Zg (cid:40) Spec A, then Spec A is reducible. Thus, we must either
have f = 0 or g = 0 and A has no zerodivisors.
Conversely, suppose t... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
k(K) is the set of k × (n − k) matrices with entries in K.
2
(cid:54)
18.725 Algebraic Geometry I
Lecture
(cid:23)
We require that this subset is open and that the isomorphism with Ak(n−k) is an isomorphism of varieties.
Notation: PV := Pn is the projectivization of V = kn (choose a basis for this).
Theorem 1.1. This ... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
2.
Finally, ω = 0 if dim ker ω = 4, then the form ω = 0.
Thus, Gr(2, 4) is isomorphic to a quadric in P5 and Gr(2, 4) ∼= Q(P5), where Q is defined by ω ∧ ω = 0.
(Exercise: Show this is an isomorphism of varieties.) Using some linear algebra, we can show that the
quadratic form is not degenerate.
For more details on work... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
��bers.
Corollary 3. If B ⊂ A and A is finitely generated over B as a B-module (“A is finite over B”), then
Spec A −→ Spec B has finite nonempty fibers.
3
18.725 Algebraic Geometry I
Lecture
(cid:23)
Proof. We only need to check that the map Spec A −→ Spec B is onto. The image is not contained in ZI
for all nonzero I ⊂ B ... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
set (nonempty since quotient ring nonzero). Thus, f has finite nonempty fibers.
Example 3. (Examples of affine morphisms)
1. Let Z ⊂ X be a closed subvariety. Then, the map i : Z (cid:44)→ X is affine and finite since Spec A/I is a
closed subset of Spec A (this is a local question). Any affine open covering of X works.
2. Let Y... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
2).
Note that f (Z1) and f (Z2) are closed by the previous lemma. We also have that the image of an
irreducible set is irreducible. This lemma shows that the images are actually distinct. We will check this
result (see Lemma 2.4.4 on p. 19 of Kempf) in the next lecture.
Definition 3. The dimension of a Noetherian topolo... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
Lecture 4
Removable Singularity Theorem
Theorem 1 Let u be harmonic in Ω \ {x0}, if
�
u(x) =
|
2−n)
o( x − x0
|
|
|
o(ln x − x0 )
, n > 2,
, n = 2
as x
→
x0, then u extends to a harmonic function in Ω.
Proof: Without loss of generality, we can assume Ω = B(0, 2), then u|∂B(0,1) is contin
uous. Thus by Poisson ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
(0) \ {0}.
By reverting u and v, we can get
v(x) ≤ u(x), ∀x ∈ B1(0) \ {0},
thus v(x) = u(x), ∀x ∈ B1(0) \ {0}.
Now we can define u(0) = v(0), and extend u to be a harmonic function on B(0, 1),
thus a harmonic function on Ω = B(0, 2).
�
Example This gives an example of Dirichlet problem that is NOT solvable:
Take Ω... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
f (r)
2
r
ΔSn−1 B(θ).
Proposition 1 Let B(θ) be a homogeneous harmonic polynomial of degree k restricted
to Sn−1, then ΔSn−1 B(θ) = −k(k + n − 2)B(θ).
Remark 1 Let Pk be the set of homogeneous polynomials of degree k on Rn , Hk be the
set of harmonic homogeneous polynomials of degree k on Rn, then
It’s not hard to ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
For p = k, we get u(r, θ) = rk B(θ), where B(θ) ∈ Hk , thus u is just the homogeneous
k harmonic polynomial on Rn .
For those p = −k − n + 2, if k = 0, then p = 2 − n and B(θ) = constant, thus
u = c · r2−n, which is the fundamental solution. if k > 0, then p < 2 − n, note that
B(θ) is defined on the compact set Sn−1... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
Engineering Systems
Doctoral Seminar
ESD.83 – Fall 2011
Class 1
Faculty: Chris Magee and Joe Sussman
TA: Rebecca Kaarina Saari
Guest: Professor Joel Moses, Institute Professor,
(EECS and ESD)
© 2010 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
1 Session 1: Overvie... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
systems
© 2009Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
3 Learning Objectives for ESD 83
Basic Literacy: Understanding of core concepts and principles – base
level of literacy on the various aspects of engineering systems
Inter-disciplinary capability: The ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
� Lecture, discussion and integration
Report from the Front
Next Week’s class (5 min.)
© 2009Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
5 Engineering Systems Doctoral Seminar,
Fall 2011
Class 1: ESD: Present and Future -The
Incorporation of Social Science a... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
of complex socio-
technical systems
Guest: TBD
© 2009 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
8 Engineering Systems Doctoral Seminar,
Fall 2011 – continued
Class 13: Sustainability
Guest: Noelle Selin
Class 14: Policy Design/Wrap-up
Guest: None
© 2009Chris Magee and ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
as part of class participation)
© 2009 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
11 Assignment Summary (syllabus) 3
5. Book Review (750 word book review, 10% of
the total)
Assigned: Session 1; Due: Session 5 (discussion during
Session 7)
6. Historical Roots and Current Metho... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
Session 1: Overview
Welcome, Overview and Introductions (10 min.)
The doctoral Seminar on Engineering Systems: Logistics
and Context (20min)
Assignments discussion and sign up process (20 min)
Discussion of C. P. Snow essay
Break (10 min.)
Discussion with Guest (Joel Moses), 40- 50 minutes)
Relationships... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
�� Are there significant differences
among different sciences?
How might one consider closeness of
relationships among sciences?
What role(s) might engineering play
in the relationships among fields?
© 2009Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
17 Cit... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
. L. Magee “Introduction”
PhD in Materials Science and Engineering
(CMU-then CIT)
Ford Research staff (1966-71, basic
research in materials)
Various Ford positions (1971-2001) mostly
in product development management)
MIT (2000- now) mostly ESD + positions
Research in complex socio-technical
systems; simu... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
Notes -on double integrals.
(Read 11.1-11.5 of Apostol.)
Just as for the case of a single integral, we have the
following condition for the existence of a double integral:
Theorem 1 (Riemann condition). Suppose f -is defined -on
Q = [arb] x [c,d]. Then f -is integrable -on Q - -if and only -if
given any E > 0, th... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
- I f
f
t h e n
o v e r - Q,
g - on Q, - -
and i f
f - and g - a r e i n t e g r a b l e
TO prove t h i s theorem, one f i r s t v e r i f i e s t h e s e r e s u l t s
f o r s t e p f u n c t i o n s (see 1 1 . 3 ) , and t h e n u s e s t h e Riemann condi-
t i o n t o prove them f o r g e n e r a l i n t e g ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
of Q relative to whi.ch all of
sl, s2, tl, t2 are step functions; then sl + s2 and tl + t2
are also step functions relative to this partition. Furthermore,
one adds the earlier inequalities to obtain
Finally, we compute
this computation uses the fact that linearity has already been
proved for step functions. ~ h ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
each fixed y - in [c,d], assume --
that the
exists. --
Then the integral
A ( y ) dy exists,
and furthermore,
,
I
1
proof. We need to show that [ X(y)dy exists and equals
I
the double integral ,b f *
Choose step functions s (x,y) and t(x,y) , defined on
Q, such that s(x,y) C f (x,y) t(x,y), and
his w e ca... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
on y j y j so it is a step func-
tion.
A similar argument shows that the function
is a step function for c -< y -< d.
Now since s 5 f 5 t for all (xfy), we have
- . .
by the comparison theorem. (T3emiddle integral exists by
hypothesis.) That is, for all y in [c,d] ,
.-
Thus S and T are step functions lying bene... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
hand, one can proceed t o c a l c u l a t e
some specific double i n t e g r a l s . S e v e r a l examples a r e worked o u t
i n 1 1 . 7 and 11.8 o f Apostol.
NOW l e t us t u r n t o t h e f i r s t of o u r b a s i c q u e s t i o n s , t h e
one concerning t h e e x i s t e n c e of t h e double i n t e g r a ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
t i j - s
s t e p f u n c t i o n s s and t w i t h s < f G t on Q. One t h e n has
E ' . U s e t h e numbers sij and t i j t o o b t a i n
<
j
i
JJ*( t - s ) < ~ ' ( d- c ) ( b - a ) .
I
i
This number e q u a l s E
E = E / (d-c) (b-a) .
i f w e begin t h e proof by s e t t i n g
I n p r a c t i c e , t h i s... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
o have c o n t e n t -z e r o i f f o r every E > 0, t h e r e i s a f i n i t e
s e t of r e c t a n g l e s whose union c o n t a i n s D
and t h e sum of whose
a r e a s does n o t exceed E .
Examples.
(I) A f i n i t e s e t h a s c o n t e n t zero.
( 2 ) A h o r i z o n t a l l i n e segment h a s c o n t e... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
v i a l t o prove: only t h e
l a s t t w o r e q u i r e some c a r e . L e t us prove ( 6 ) . Let
E ' > 0 .
Given t h e continuous function , l e t us use t h e small-
span theorem f o r functions of a s i n g l e v a r i a b l e t o choose a
p a r t i t i o n a = xo < x l < ... < xn. = b of
[a ,b] such t h a t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
n t e n t zero:
Lemma 5. - L e t Q - -
be a r e c t a n g l e . - L e t D - -
be a s u b s e t - o f
i s a p a r t i t i o n - of
t h a t has c o n t e n t - zero. Given E > 0 ,
Q --
such t h a t t h o s e s u b r e c t a n g l e s - -
Q --
p o i n t s - o f D - have t o t a l ---
a r e a l e s s t h a n E.
... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
i g u r e i l l u s t r a t e s t h e d i s t i n c t i o n ; D
i s l e s s
i s c o n t a i n e d i n t h e union of two s u b r e c t a n g l e s , b u t t h e r e a r e
i
seven s u b r e c t a n g l e s t h a t c o n t a i n p o i n t s of D.
Proof. F i r s t , choose f i n i t e l y many r e c t a n g l e s
A1,... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
may extend
o u t s i d e Q ,
s o l e t A:
denote t h e r e c t a n g l e t h a t i s t h e
i n t e r s e c t i o n of A:
and Q. Then t h e r e c t a n g l e s A:
a l s o
have t o t a l a r e a l e s s t h a n E .
Now use t h e end p o i n t s of t h e component i n t e r v a l s of t h e
r e c t a n g l e s AT... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
t of D ,
it c o n t a i n s a p o i n t o f
l i e s i n % and hence i n Ai;.
Suppose w e l e t B d e n o t e t h e union
o f a l l t h e s u b r e c t a n g l e s Q i j
t h a t c o n t a i n p o i n t s of D ;
and l e t
A be the union of t h e r e c t a n g l e s A
A Then B C A.
It follows that
1 area Qij
Q~... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
5 tl and s2 5 h j t2,
and such that
Consider the step functions sl and t2. We know that
s 1 I g l f 1 h 1 t 2
so sl is beneath f, and t2 is above f. Furthermore, because the integral of g is between
the integrals of sl and of tl, we know that
Similarly,
If we add these inequalities and the inequality
we have
S... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
rectangle Q.. that does contain a point of D, because it is a step function on such a
1J
subrectangle. (It is constant on the interior of Q. ..) The additivity property of the
U
integral now implies that g is integrable over Q.
Similarly, h is integrable over Q. Using additivity, we compute the integral
JLQ (h-g)... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
1J
JJQ
h = M (1(area Q.. that contain points of D))
13
Similarly,
E, SO that f is integrable over Q. Furthermore,
Since E is arbitrary,
Corollary 8. aIJQf exists, and if g g bounded function that eauds f except on a
set of content
--
\
1
P r o o f .
We w r i t e g = f
+
( g - f ) . Now
f
i s i n t e g ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
e g e n e r a l c a s e .
F i r s t , w e prove t h e f o l l o w i n g b a s i c e x i s t e n c e theorem:
Theorem 9. - L e t S be - - a bounded s e t i n t h e p l a n e .
I f
J-Mm,EZ-bn.slLYUll -
Bd S -h a s c o n t e n t -zero, - -and i f f -1 i s c o n t i n u o u s --a t each p o i n t
-of
e x i s t s .
b... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
s where
can' fail t o b e continuous
are p o i n t s of ths boundary of S , and this set, by assumption,
has c o n t e n t zero. Hence f!
e x i s t s . El
Q
-Note: Adjoining or deleting boundary points of S
changes the value of f only on a set of content zerol so that
value of /Js f remains unchanged. Thus )'I... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
right side exists.
Proof. (a) Given f, g defined on S, let 2, g equal
f t gr respectively, on S and equal 0 otherwise. Then
cl + dg equals cf + dg on S and 0 otherwise. Let Q
be a rectangle containing S. We know that
t
from this linearity follows.
(b) Similarly, if f -
< g, then
,
from which we
conclude that ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
is of type I
and S2 is of type 11, we can compute the integrals I[
and {I
s,
f by iterated integration. We add the results to
s,A.
f
obtain
-Area.
We can now construct a rigorous theory of area. We
already have defined the area of the rectangle Q = [a,b] x [c,d]
by the equation
area Q = 11 I..
Q
,I
We use ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
S) .
Proof. Let Q be a rectangle containing S and T.
Let
is(x) = 1 for x E S
= 0 for x ft S.
Define FT similarly.
(1) If S is contained in T, then $ (x) C lT(x) .
Then by the comparison theorem,
area s = Ifs 1 = 11, L < /I,%
= j'b I. = area T.
(2) Since 0 < 1, we have by the comparison theorem,
0 = 11, 0. 11... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
$1 1 exists
andequals $1 1 + $ $ 1.
SUT
S
T
(4) Since the part of S not in Int S lies in Bd S,
it has content zero. Then additivity implies that
area S = area(1nt S) + area (S - Int S)
= area (Int S) .
A similar remark shows that
area ( S u Bd SJ = area (Int S ) + area(Bd S)
= area (Int S ) .
1
Remark.
L... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
S
be t h e supremum
t h a t
ranges o v e r a l l p a r t i t i o n s o f Q;
of t h e numbers a ( P ) , a s P
and d e f i n e t h e o u t e r -a r e a o f S t o b e t h e infemum of t h e numbers
A ( P ) . If t h e i n n e r a r e a and o u t e r a r e a of S a r e e q u a l , t h e i r
common v a l u e i s c a l ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
v a r i a n c e o f a r e a under "congruence" w i l l
have t o w a i t u n t i l we s t u d y t h e problem o f change of v a r i a b l e s
i n a double i n t e g r a l .
I
Eh ercises
1. Show t h a t i f ISS 1 e x i s t s , then Bd S hz-s content zero.
[Hint: Chwse Q s o t h a t S C Q . Since S& IS e x i s t s ,... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
5. Express i n t e r n of i t e r a t e d i n t e g r a l s t h e volume of t h e region
i n t h e f i r s t a t a n t of R~ bcunded by:
(a) The s u r f a c e r z = xy and
z = 0 w d x + 2y + z = 1.
( b ) The s u r f a c e s z = xy and z = 0 and
Let Q denote the rectangle [0,1] x [0,1] in the following exercises. ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
T Y
R
E
B
I
L
4
199
Source: Wikipedia
Jacob (James) Bernoulli
(1654–1705)
For Bernoulli Trials
Figure by MIT OCW.
ESD.86
Class #2
February 12, 2007
Analyzing a Probability Problem
Four Steps to Happiness
1. Define the Random Variable(s)
2. Identify the (joint) sample space
3. Determine the probability law over the
... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
over the sample space?
Binomial Probability Mass Function
How many people go by you until you
have completed your kth interview?
Let Y=number of people who pass by up to and
including the one who is the kth person interviewed.
P{Y=y}=P{exactly k-1 interviews occur in (y-1)
people passing AND the yth person passing ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
into a closet upon arrival.
The host, at the end of the party, gives a random hat to each
departing player. What is the expected number of hats that are
returned to their rightful owners?
Solution: Define indicator r.v.
Let H = the number of hats returned to the correct owners. Then,
Answer independent of the number o... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
nth flip is the 1st Heads, you win $2n. The bank is
Donald Trump, so this game can go on for a long
long time!
(a) How much would you be willing to pay for
playing this game?
(b) What is the expected dollar value of winnings in
the game?
This is the St.Petersburg Paradox
Daniel Bernoulli (1738; English trans. 19... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
8 kg (392 lb)
1 metric ton
5.6 metric tons
32 metric tons
178 metric tons
1 kiloton
5.6 kilotons
32 kiloton
178 kilotons
1 megaton
5.6 megatons
50 megatons
178 megatons
1 gigaton
5.6 gigatons
32 gigatons
0.5
1.0
1.5
2.0
2.5
4.0
3.0
3.5
4.5
5.0
Hand grenade
Construction site blast
WWII conventional
bombs
late WWII conve... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
and B}=P{AB}= pA pB
Add one redundant arc to increase Reliability
A
1
B
C
2
Let T12 = event successful Transmission from node 1 to node 2.
pA=P{Link A works properly}=P{A}
pB= P{Link B works properly}=P{B}
pC= P{Link C works properly}=P{C}
Assume links A, B and C function independently.
Then,
P{T12}=P{A and (B or C)}... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
%jacobian2by2.m
%Code 8.1 of Random Eigenvalues by Alan Edelman
%Experiment: Compute the Jacobian of a 2x2 matrix function
%Comment: Symbolic tools are not perfect. The author
% exercised care in choosing the variables.
syms p q r s a b c d t e1 e2
X=[p q ; r s]; A=[a b;c d];
%% Compute Jacobians
Y=X^2; J=jacob... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/2b7145efcda0c3575006b63814358991_jacobian2by2.pdf |
Intro
Administrivia.
• Signup sheet.
• prerequisites: 6.046, 6.041/2, ability to do proofs
• homework weekly (first next week)
•
collaboration
• independent homeworks
• grading requirement
• term project
•
books.
• question: scribing?
Randomized algorithms: make random choices during run. Main benefits:
• speed... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
)
– hashing to same buckets
– online algorithms
– note: “adversarial” may mean “well structured” i.e. natural
• fingerprinting/verification
– generate short random fingerprints for things
– faster than comparing things
– almost every fingerprint works
– so a random one works
2
• random sampling. graph algs, comput... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
. . . , Sj . all equally likely
• Si and Sj get compared if pivot is Si or Sj
• probability is at most 2/(j − i + 1) (may have outer elements)
• analysis:
n
n
pij ≤
�
�
i=1 j>i
2/(j − i + 1)
�
�
i=1 j>i
n n−i+1
2/k
1/k
=
�
�
i=1 k=1
n
n
≤ 2
�
�
i=1 k=1
≤ 2nHn
4
(Define Hn, claim O(log n)... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
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