In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Bayesian network structure estimation based on the Bayesian/MDL criteria when...Joe Suzuki
J. Suzuki. ``Bayesian network structure estimation based on the Bayesian/MDL criteria when both discrete and continuous variables are present". IEEE Data Compression Conference, pp. 307-316, Snowbird, Utah, April 2012.
Optimal interval clustering: Application to Bregman clustering and statistica...Frank Nielsen
We present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means, k-medoids, k-medians, k-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate k by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.
http://arxiv.org/abs/1403.2485
Accelerated reconstruction of a compressively sampled data streamPantelis Sopasakis
Recursive compressed sensing on a stream of data: The traditional compressed sensing approach is naturally offline, in that it amounts to sparsely sampling and reconstructing a given dataset. Recently, an online algorithm for performing compressed sensing on streaming data was proposed: the scheme uses recursive sampling of the input stream and recursive decompression to accurately estimate stream entries from the acquired noisy measurements.
In this paper, we develop a novel Newton-type forward-backward proximal method to recursively solve the regularized Least-Squares problem (LASSO) online. We establish global convergence of our method as well as a local quadratic convergence rate. Our simulations show a substantial speed-up over the state of the art which may render the proposed method suitable for applications with stringent real-time constraints.
Fast parallelizable scenario-based stochastic optimizationPantelis Sopasakis
Fast parallelizable scenario-based stochastic optimization: a forward-backward LBFGS method for stochastic optimal control problems with global convergence rate guarantees. (Talk at EUCCO 2016, Leuven, Belgium).
Forest Learning based on the Chow-Liu Algorithm and its Application to Genom...Joe Suzuki
J. Suzuki, ``Forest Learning based on the Chow-Liu Algorithm and its Application to Genome Differential Analysis: A Novel Mutual Information Estimation", AMBN 2015, Yokohama, Japan
Structure Learning of Bayesian Networks with p Nodes from n Samples when n<...Joe Suzuki
``Structure Learning of Bayesian Networks with p Nodes from n Samples when n<<p">, presented at Probabilistic Graphical Model Workshop, ISM, March 2016.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
1. .
......
Bayesian Criteria based on Universal Measures
Joe Suzuki
Osaka University
October 29, 2012
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 1 / 18
2. Road Map
...1 Problem
...2 Density Functions
...3 Generalized Density Functions
...4 The Bayesian Solution
...5 Summary
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 2 / 18
3. Problem
Warming-Up
Identify whether X, Y are independent or not, from n examples
(x1, y1), · · · , (xn, yn) ∼ (X, Y ) ∈ {0, 1} × {0, 1}
p: a prior probability that X, Y are independent
.
The Bayesian answer
..
......
Consider some weight W to compute
Qn
(xn
) :=
∫
P(xn
|θ)dW (θ) , Qn
(yn
) :=
∫
P(yn
|θ)dW (θ)
Qn
(xn
, yn
) :=
∫
P(xn
, yn
|θ)dW (θ)
pQn(xn)Qn(yn) ≥ (1 − p)Qn(xn, yn) ⇐⇒ X, Y are independent
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 3 / 18
4. Problem
Today’s Exercise
A similar problem but what if (X, Y ) ∈ [0, 1) × {1, 2, · · · }.
.
Problem
..
......Construct something like Qn(xn), Qn(yn), Qn(xn, yn).
Extend the idea without assuming either discrete or continuous
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 4 / 18
5. Problem
What Qn
is qualified to be an alternative to Pn
?
θ∗: true θ
Pn(xn) = P(xn|θ∗), Pn(yn) = P(yn|θ∗) Pn(xn, yn) = Pn(xn, yn|θ)
Qn
(xn
) :=
∫
P(xn
|θ)dW (θ) , Qn
(yn
) :=
∫
P(yn
|θ)dW (θ)
Qn
(xn
, yn
) :=
∫
P(xn
, yn
|θ)dW (θ)
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 5 / 18
6. Problem
Example: Bayes Codes
c: the # of ones in xn
P(xn
|θ) = θc
(1 − θ)n−c
a > 0
w(θ) ∝
1
θa(1 − θ)a
For each xn = (x1, · · · , xn) ∈ {0, 1}n,
Qn
(xn
) :=
∫
w(θ)P(xn
|θ)dθ
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 6 / 18
7. Problem
Universal Coding/Measures
If we choose
a = 1/2
(Krichevsky-Trofimov) and xn is i.i.d. emitted by
Pn
(xn
) =
n∏
i=1
P(xi )
then, for any P, almost surely,
−
1
n
log Qn
(xn
) → H :=
∑
x∈A
−P(x) log P(x)
From Shannon McMillian Breiman, for any P,
−
1
n
log Pn
(xn
) =
1
n
n∑
i=1
− log P(xi ) → E[− log P(xi )] = H
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 7 / 18
8. Problem
The Essential Problem
For any P, almost surely,
1
n
log
Pn(xn)
Qn(xn)
→ 0 (1)
(explains why Pn can be replaced by Qn if n is large)
.
X is neither discrete nor continuous
..
......What are Qn and (1) in the general settings ?
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 8 / 18
9. Density Functions
Suppose a density function exists for X
A: the range of X
A0 := {A}
Aj+1 is a refinement of Aj
Example 1: if A0 = {[0, 1)}, the sequence can be
A1 = {[0, 1/2), [1/2, 1)}
A2 = {[0, 1/4), [1/4, 1/2), [1/2, 3/4), [3/4, 1)}
. . .
Aj = {[0, 2−(j−1)), [2−(j−1), 2 · 2−(j−1)), · · · , [(2j−1 − 1)2−(j−1), 1)}
. . .
sj : A → Aj (projection, x ∈ a ∈ Aj =⇒ sj (x) = a)
λ : R → B (Lebesgue measure, a = [b, c) =⇒ λ(a) = c − b)
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 9 / 18
10. Density Functions
If (sj (x1), · · · , sj (xn)) = (a1, · · · , an),
gn
j (xn
) :=
Qn
j (a1, · · · , an)
λ(a1) · · · λ(an)
f n
j (xn
) := fj (x1) · · · fj (xn) =
Pj (a1) · · · Pj (an)
λ(a1) . . . λ(an)
For {ωj }∞
j=1:
∑
ωj = 1, ωj > 0, gn
(xn
) :=
∞∑
j=1
ωj gn
j (xn
)
If we choose {Ak} such that fk → f , for any f , almost surely
1
n
log
f n(xn)
gn(xn)
→ 0 (2)
B. Ryabko. IEEE Trans. on Inform. Theory, 55, 9, 2009.
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 10 / 18
11. Generalized Density Functions
Exactly when does density function exist?
B: the Borel sets of R
µ(D): the probabbility of D ∈ B
.
When a density function exists
..
......
The following are equivalent (µ ≪ λ):
for each D ∈ B, λ(D) = 0 =⇒ µ(D) = 0
∃ B-measurable
dµ
dλ
:= f s.t. µ(D) =
∫
D
f (t)dλ(t)
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 11 / 18
12. Generalized Density Functions
Density Functions in a General Sense
.
Radon-Nikodum’s Theorem
..
......
The following are equivalent (µ ≪ η):
for each D ∈ B, η(D) = 0 =⇒ µ(D) = 0
∃ B-measurable
dµ
dη
:= f s.t. µ(D) =
∫
D
f (t)dη(t)
Example 2: µ({k}) > 0, η({j}) :=
1
k(k + 1)
, k ∈ B := {1, 2, · · · }
µ ≪ η
µ(D) =
∑
k∈D∩B
f (k)η({k})
dµ
dη
(k) = f (k) =
µ({k})
η({k})
= k(k + 1)µ({k})
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 12 / 18
13. Generalized Density Functions
In this work, ...
B1 := {{1}, {2, 3, · · · }}
B2 := {{1}, {2}, {3, 4, · · · }}
. . .
Bk := {{1}, {2}, · · · , {k}, {k + 1, k + 2, · · · }}
. . .
tk : B → Bk (projection, y ∈ b ∈ Bk =⇒ tk(y) = b)
If (tk(y1), · · · , tk(yn)) = (b1, · · · , bn),
gn
k (yn
) :=
Qn
k (b1, · · · , bn)
η(b1) · · · η(bn)
, gn
(yn
) :=
∞∑
k=1
ωkgn
k (yn
)
If we choose {Bk} s.t. fk → f , for any f , almost surely
1
n
log
f n(yn)
gn(yn)
→ 0 (3)
gn(yn)
∏n
i=1 ηn({yi }) estimates P(yn) = f n(yn)
∏n
i=1 ηn({yi })
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 13 / 18
14. Generalized Density Functions
Joint Density Functions
Example 3: A × B (based on Examples 1,2)
µ ≪ λη
A0 × B0 = {A} × {B} = {[0, 1)} × {{1, 2, · · · }}
A1 × B1
A2 × B2
. . .
Aj × Bk
. . .
(sj , tk) : A × B → Aj × Bk
If {Aj × Bk} satisfies fjk → f , for any f , almost surely, we can construct
gn s.t.
1
n
log
f n(xn, yn)
gn(xn, yn)
→ 0 (4)
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 14 / 18
15. The Bayesian Solution
The Answer to Today’s Problem
Estimate f n
X (xn), f n
Y (yn), f n
XY (xn, yn) by
gn
X (xn), gn
Y (yn), gn
XY (xn, yn)
.
The Bayesian answer
..
......pgn
X (xn)gn
Y (yn) ≤ (1 − p)gXY (xn, yn) ⇐⇒ X, Y are independent
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 15 / 18
16. The Bayesian Solution
The General Bayesian Solution
Givem n example zn and prior {pm} over models m = 1, 2, · · · ,
compute gn(zn|m) for each m = 1, 2, · · ·
find the model m maxmizing pmg(zn|m)
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 16 / 18
17. The Bayesian Solution
Universality in the generalized sense
1
n
log
f n(zn)
gn(zn)
→ 0
µn
(Dn
) :=
∫
D
f n
(zn
)dηn
(zn
)
νn
(Dn
) :=
∫
D
gn
(zn
)dηn
(zn
)
f n(zn)
gn(zn)
=
dµn
dηn
(zn
)/
dνn
dηn
(zn
) =
dµn
dνn
(zn
)
.
Universality
..
......
1
n
log
dµn
dνn
(zn
) → 0
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 17 / 18
18. Summary
Summary and Discussion
.
Bayesian Measure
..
......
Generalization without assuming Discrete or Continuous
Universality of Bayes/MDL in the generalized sense
.
Many Applications
..
......
Bayesian network structure estimation (DCC 2012)
The Bayesian Chow-Liu Algorithm (PGM 2012)
Markov order estimation even when {Xi } is continuous
Joe Suzuki (Osaka University) Bayesian Criteria based on Universal Measures October 29, 2012 18 / 18