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2abd5d121e6ed321961e08edca4ba0d457022210 | subsection | 151 | 167 | Area Queries | This further implies that for
a leaf box b, the quantity |{3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} \cup {3far}{1}{U_{b}}{{3f... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.0011943118879571557,
0.026603011414408684... |
e4e0f5df63b5921444e7b055424dcae62df6169c | subsection | 152 | 167 | Area Queries | Next, we show that {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} \subseteq {4}{1}{U_{b}}{{4}{2}{V_{b}}{{4}{3}{W_{b}}{{4}{3close}{... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
ca4763f38d36bfdd199034c1f9d55464002fa762 | subsection | 153 | 167 | Area Queries | If b^{\prime } is an ancestor of b, then b must be separated by
an \ell ^\infty distance of 2^{k+1}|b| from any box in {4}{1}{U_{b^{\prime }}}{{4}{2}{V_{b^{\prime }}}{{4}{3}{W_{b^{\prime }}}{{4}{3close}{W^\mathrm {close}_{b^{\prime }}}{{4}{3far}{W^\mathrm {far}_{b^{\prime }}}{{4}{4}{X_{b^{\prime }}}{{4}{4close}{X^\math... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0... |
d5b6dcc0ee91b2e34d380454307ee0730c514094 | subsection | 154 | 167 | Area Queries | Finally, |{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}{{4far}{3close}{W^\mathrm {close}_{b}}{{4far}{3far}{W^\mathrm {far}_{b}}{{4far}{4}{X_{b}}{{4far}{4close}{X^\mathrm {close}_{b}}{{4far}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}| \le 375 follows since, by definition,
{4far}{1}{U_{b}}{{4far}{2}{V_{b}}{{4far}{3}{W_{b}}... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
87e1f3ef740917815131a09f766c6ed50ba4233e | subsection | 155 | 167 | Numerical Experiments in Support of FMM Translation Error Estimates | The code used for the numerical experiments performed to obtain the results of
Section is available
at . In this appendix, we describe the
procedure the code uses. | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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9fe456440c0f4403feb82cfc8f026e02c48f7437 | subsection | 156 | 167 | Multipole and Multipole-to-Local Accuracy | We use the notation of
Section REF . Hypothesis REF pertains to the
accuracy of approximating a local expansion using an intermediate multipole
expansion.As a numerical experiment, we test the truth of this hypothesis at selected
values of the parameters (R, r, \rho , p, q). For a given value of these
parameters, estim... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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a6fd9c0f8098026278034ea0f37f248b15a92ade | subsection | 157 | 167 | Detailed Complexity Analysis | This section provides the details of the complexity analysis from
Section REF , under the assumptions highlighted in
Section REF . In
Section REF and REF we
review the complexity of translations and the effect of the particle
distribution. Section REF provides the details
supporting the analysis in Table REF .We use th... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
9569f7b8c33284f88d66b5308f6228d1b5f158d4 | subsection | 158 | 167 | Complexity of Translation Operators | A p-th order multipole/local expansion requires (p+1)^2 expansion
coefficients. The (p+1)^2 corresponding basis functions for the coefficients
may be evaluated in O(p^2) time using well-known
recurrences . As a result, we model the cost of forming
or evaluating a p-th order multipole/local expansion in spherical harmon... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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6e58e4d253fa277a5159e90e997552520689f8d6 | subsection | 159 | 167 | Effect of Particle Distribution | The running time of the GIGAQBX FMM cannot be entirely independent of the
particle distribution. Unlike the point FMM, the algorithm may place more than
{n_{\mathrm {max}}} particles in a box. This occurs due to clustering of QBX centers in
boxes because of the target confinement rule. This phenomenon cannot be
disrega... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
b5b406906bf82e5c9829e79317f6a3c129efdc87 | subsection | 160 | 167 | Effect of Particle Distribution | The following proposition is an
immediate consequence of the definition of M_C and the previous lemma.Proposition 9 (Bound on 2-near-neighborhood interactions for QBX expansions) The number of source-center pairs (s, c), such that c is a suspended QBX center and
s is a source particle in the 2-neighborhood of the box o... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0... |
b75d193b212a7145dd8313751b1129fe98eaa30c | subsection | 161 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | Proposition 10 (Number of larger leaf boxes in the 1-neighborhood of a box) Let b be a box. There are at most 27 leaf boxes at least at large as b
intersecting the 1-near neighborhood of b.Let l be such a leaf box. If l \ne b, choose a box c_l which is a
colleague of b that is geometrically contained inside l. The mapp... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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5a65f7352771f6e62a0d24fd463e19ec98d945ae | subsection | 162 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | This assumption will not lead to an
undercount of the cost of the Stage 5 interactions, if the optimization in
Remark REF has been applied. Recall that a List 3
far interaction is a multipole-to-target interaction, and a List 3 close
interaction is a source-to-target interaction. The only way the above
assumption could... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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79ebd23b65c727630718b8b6950b7addf7f28985 | subsection | 163 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | If the tree is level-restricted, this implies that if
b is a leaf box, any box in {3close}{1}{U_{b}}{{3close}{2}{V_{b}}{{3close}{3}{W_{b}}{{3close}{3close}{W^\mathrm {close}_{b}}{{3close}{3far}{W^\mathrm {far}_{b}}{{3close}{4}{X_{b}}{{3close}{4close}{X^\mathrm {close}_{b}}{{3close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} \... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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8f082e28ec7134908231645e3b818b2035dd8e35 | subsection | 164 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | Every box in X_b is a leaf that is
either a 2-colleague of b not adjacent to b, or adjacent to the parent
of b and at least as large as the parent of b. There are at most 5^3 -
3^3 = 98 boxes that fall into the first category and, by
Proposition REF , at most 27 boxes that fall
into the second category.Next, we show th... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
7b9fd96bad73a38b14ae39bc38d3f920f74439e4 | subsection | 165 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | Recall that {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}} must be a subset of the List 4's of the
ancestors of b. If b^{\prime } ... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0... |
4a1b58e5205fadbc4182d560b8e33ca1502b1b88 | subsection | 166 | 167 | Complexity of Algorithmic Stages Associated with Interaction Lists | It follows that {4close}{1}{U_{b}}{{4close}{2}{V_{b}}{{4close}{3}{W_{b}}{{4close}{3close}{W^\mathrm {close}_{b}}{{4close}{3far}{W^\mathrm {far}_{b}}{{4close}{4}{X_{b}}{{4close}{4close}{X^\mathrm {close}_{b}}{{4close}{4far}{X^\mathrm {far}_{b}}{}}}}}}}}
is disjoint from the List 4 of a grandparent of b or above.Finally,... | {
"cite_spans": []
} | 10.1016/j.jcp.2019.03.024 | 1805.06106 | A Fast Algorithm for Quadrature by Expansion in Three Dimensions | [
"Matt Wala",
"Andreas Klöckner"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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277898a947a281c6c89ea3feb0dc899c3e474d8c | abstract | 0 | 108 | Abstract | When smoothing a function $f$ via convolution with some kernel, it is often
desirable to adapt the amount of smoothing locally to the variation of $f$. For
this purpose, the constant smoothing coefficient of regular convolutions needs
to be replaced by an adaptation function $\mu$. This function is matrix-valued
which ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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b1c9d889e084167bd48bbe50d9eebb6124b05e67 | subsection | 1 | 108 | Motivation | The convolution of two integrable functions f,g\colon \mathbb {R}^d\rightarrow \mathbb {R},(f\ast g)(x) = \int _{\mathbb {R}^d}f(y)\, g(x-y)\, \mathrm {d}y,is a basic mathematical tool with applications in probability theory, image
processing, optics, acoustics and many other areas.
If g is a probability density, say a... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
d62dcc210aef81e178b2c1f3b6e6c6601df1caff | subsection | 2 | 108 | Motivation | However, if the variation of the function changes considerably in space, no
single suitable width \sigma can be found and one is forced to adapt it locally,(f\ast _{\mu } g) (x)\mathrel {\mathop }= \int f(y)\, *{\det \mu (y)}\, g\big (\mu (y)(x-y)\big )\,
\mathrm {d}y\, ,where \mu \mathrel {\mathop }\mathbb {R}^d\right... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1805.01729",
"end": 1972,
"openalex_id": "https://openalex.org/W2802336905",
"raw": "I. Klebanov. Axiomatic approach to variable kernel density estimation. ArXiv e-prints, 2018.",
"source_ref_id": "5122feebb607cdd56ab37... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
3c01b94a62cb22c0e156966367cda8ad0cf9b70f | subsection | 3 | 108 | Theoretical Properties of Adaptive Convolutions | Definition 3 (adaptive convolutions, adaptation function)
We define the generalized convolution f\bar{\ast }G of two
measurable functions f\colon \mathbb {R}^d\rightarrow \mathbb {R} and G \colon \mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R} as
the integral operator with kernel G:(f\bar{\ast }G)(x) \mathre... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1098/rspa.1912.0086",
"end": 2257,
"openalex_id": "https://openalex.org/W1970763581",
"raw": "W. Young. On the multiplication of successions of fourier constants. Proceedings of the Royal Society of London. Series A, Containing Papers of... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0012041839072480798,
... | |
8e4f13216b94aa47ad2ccb5d596454c753f28b4d | subsection | 4 | 108 | Theoretical Properties of Adaptive Convolutions | This will result in the generalization of
Young's inequality for convolutions to the case of adaptive convolutions.
Theorem 5
Let f\in L^1(\mathbb {R}^d), 1\le p\le \infty and G \colon \mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R} be
measurable such that \left\Vert G({\cdot ,y) _p\le \Gamma for some \Gamma ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0031438947189599276,
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-0.03333139047026634,
-0.024861188605427742,
... | |
87afdca6d569c60251da044a44139f4134742897 | subsection | 5 | 108 | Theoretical Properties of Adaptive Convolutions | Then
\Vert f\bar{\ast }G\Vert _{r}\le \Vert f\Vert _{q}\, \Gamma \, .
}
\right.}\begin{}
In the particular case p=1 we also have
\begin{align*}
\int _{\mathbb {R}^d} \left(f\ast _{\mu }g\right)(x) \, \mathrm {d}x
&=\int _{\mathbb {R}^d} f(y) \int _{\mathbb {R}^d} g_\mu (x,y)\, \mathrm {d}x \, \mathrm {d}y
=
\left(\in... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.00937555730342865,
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0.0018222832586616278,
-0.03069751150906086,
... | |
8176d86712133ad193eedeb973c29dbcbc91079b | subsection | 6 | 108 | Theoretical Properties of Adaptive Convolutions | Further, k vectors v_1,\dots ,v_k\in \mathbb {R}^d give rise to a linear form on the space
\mathcal {SML}^k\left(\mathbb {R}^d\right) of
k-fold symmetric multilinear forms on \mathbb {R}^d,
[v_1,\dots ,v_k]\colon \mathcal {SML}^k\left(\mathbb {R}^d\right) \rightarrow \mathbb {R},\qquad \phi \mapsto \phi \left(v_1,\dot... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03481641039252281,
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0.004256695043295622,
0.0016563350800424814,
-0.0... | |
0e4b1c6bd7161ef7813b0de55a22c96497829a60 | subsection | 7 | 108 | Theoretical Properties of Adaptive Convolutions | Then f\ast _{\mu }^p g\in C^m(\mathbb {R}^d) and for all \alpha \in \mathbb {N}^d with
|\alpha |\le m, the derivative \partial ^{\alpha }\left(f\ast _{\mu }^p
g\right)\in L^p(\mathbb {R}^d) is given by (we slightly abuse the notation as mentioned in Remark REF (f))
\partial ^{\alpha }\left(f\ast _{\mu }^p g\right)
=
\... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.02716755121946335,
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-0.011547353118658066,
-0.02182861603796482... | |
b5f3f3b9aee881e0fa601e75420b2ef678e7cf02 | subsection | 8 | 108 | Theoretical Properties of Adaptive Convolutions | Further, let g\in L^p\cap C^1(\mathbb {R}^d) for some 1\le p <\infty ,
\gamma (x) \mathrel {\mathop }=
x\, g(x)
,\qquad N_t(x) =
\begin{pmatrix}
j_t(x)^{\intercal } \left(D_x
\left(\mu _{t}\right)_{1,{\cdot }^{\intercal }^{\intercal }\hspace{-2.84544pt}(x)\,
}\right.\\ \vdots \\
j_t(x)^{\intercal } \left(D_x
\left(\mu... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0027620543260127306,
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-0.044955868273973465,
0.007935184054076672,
... | |
09b37495907748fb5b8e0fabafb0931e8f655aa8 | subsection | 9 | 108 | Theoretical Properties of Adaptive Convolutions | \end{pmatrix}
Corollary 8 Under the assumptions of Proposition REF , if
\rho _{g,t} = \rho _t\ast g_{A_t}
is the common convolution, but with time-dependent scaling matrix
\left(A_t\right)_{t\in \mathbb {R}} \in C^2\left(\mathbb {R},\mathrm {GL}(d,\mathbb {R})\right) of the smoothing
kernel,
g_{A_t}(x) = |\det A_t|\,... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.028990963473916054,
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... | |
ec281849b0eb481f55dea2bcc53fef18726b1fad | subsection | 10 | 108 | Theoretical Properties of Adaptive Convolutions | Remark 9 The square root M^{1/2} = \sqrt{M} of a symmetric and positive definite matrix
M\in \mathbb {R}^{d\times d} will denote the unique symmetric and positive definite matrix N\in \mathbb {R}^{d\times d} such that N^2 = M (see ).
Let us first gather some conditions, which we would like our adaptation function
\mu ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 233,
"openalex_id": "https://openalex.org/W2610857016",
"raw": "R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, 2012.",
"source_ref_id": "6e1fd94cd82860a43a9f3434762507741c7ba0cf",
"start": 0... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.004726010840386152,
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-0.01838528923690319,
-0.0077164811082184315... | |
be5a88870390dbe8f1b32a26364f2da01af0fbe8 | subsection | 11 | 108 | Theoretical Properties of Adaptive Convolutions | Axiom (A4) guarantees that a sum f = \sum _{k=1}^K f_k
of several functions f_1,\dots ,f_K with `far apart^{\prime } supports is smoothed in
approximately the same way as these functions would have been smoothed
separately, f\ast _{\mu _f} g \approx \sum _{k=1}^K f_k\ast _{\mu _{f_k}}g, see
also the motivating Example ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.03348110616207123,
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0.014665151946246624,
-0.017930855974555016,
... | |
dcb6fad6b52ad4ca10bf5f82c44fdd4c52db24a6 | subsection | 12 | 108 | Theoretical Properties of Adaptive Convolutions | Roughly speaking, covariance
matrices are easier to treat than their square roots.
}
\end{}\end{}}One possible choice that fulfills the Adaptation Axioms
\ref {cond\mathrel {\mathop }adaptation} (A1)--(A4) is
\mu _f^{(a)} = \sqrt{\frac{\nabla f\, \nabla f^{\intercal }}{f^2}}.
However, \nabla f\nabla f^{\intercal } is... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.014131917618215084,
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0.0012552377302199602,
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-... | |
468f57e2ce6fa8da3656e539c49b2d53ec359b3f | subsection | 13 | 108 | Theoretical Properties of Adaptive Convolutions | Q\in \mathrm {GL}(d,\mathbb {R}), Q^{\intercal } = Q,
the adaptive windowed Fourier transform \mathcal {F}_{Q}\mathrel {\mathop }
\mathcal {S}(\mathbb {R}^d,\mathbb {C})\rightarrow \mathcal {S}(\mathbb {R}^d\times \mathbb {R}^d,\mathbb {C}) with variable
window width Q\colon \mathbb {R}^d\rightarrow \mathrm {GL}(d,\ma... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.031242098659276962,
0.001943092211149633,
0.02... | |
6f29618edaa7950e18afc948b1fb07734171d6f7 | subsection | 14 | 108 | Theoretical Properties of Adaptive Convolutions | Proposition 12 (Plancherel Theorem and Fourier Inversion Formula)
The Fourier transform is an isometric isomorphism on \mathcal {S}(\mathbb {R}^d,\mathbb {C}) with
inverse given by
\mathcal {F}^{-1} f(x) = (2\pi )^{-d/2} \int _{\mathbb {R}^d} f(\xi )\,
e^{ix^\intercal \xi }\, \mathrm {d}\xi \, .
See .
For f\in \mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.032735612243413925,
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0.017527129501104355,
0.01838136650621891,
0.... | |
0d6174b9aed4e1fadb1825397df1a76de119bada | subsection | 15 | 108 | Theoretical Properties of Adaptive Convolutions | This issue is a manifestation of the so-called uncertainty principle, see the
discussion in \cite [Chapter 2]{grochenig2001foundations}.
}The adaptive windowed Fourier transform (\ref {equ\mathrel {\mathop }adaptiveWindowedFourier})
allows to perform this trade-off differently in different regions of \mathbb {R}^d by
c... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.008751812390983105,
0.007485203444957733,
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0.0027392334304749966,
-0.02765176072716713,
... | |
a42deb01b80464144016c4e315cfed3ddaf2c3a3 | subsection | 16 | 108 | Theoretical Properties of Adaptive Convolutions | The spectral density also has the proper behaviour under scaling of f – if
f is scaled by some factor \alpha \ne 0, \tilde{f}(x) = f(\alpha x), the
(global) variation is scaled by \alpha ^{-1} and in fact the Fourier transform
(and thereby the spectral density) is scaled accordingly:
\mathcal {F}\tilde{f} (\xi )
=
(2... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.0007025151862762868,
0.004849405959248543,
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0.0418781079351902,
-0.034216735512018204,
0.03580395132303238,
-0.014704649336636066,
-0.000... | |
065783bc2a8f2f3b533ec27ddb67f96613b91556 | subsection | 17 | 108 | Theoretical Properties of Adaptive Convolutions | Again, let us consider the expectation value and covariance of the corresponding
probability density in \xi :
Proposition 14
Let f\in W^{2,2}(\mathbb {R}^d,\mathbb {R})\setminus \lbrace 0\rbrace and Q\in \mathrm {GL}(d,\mathbb {R}), Q^{\intercal } = Q.
Then, for each x\in \mathbb {R}^d, the expectation value and covar... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03192894533276558,
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0.004021642729640007,
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0.0286475270986557,
-0.0199632216244936,
0.02... | |
8f83262af6cbf363ff88a48190d9cd14ae5f8273 | subsection | 18 | 108 | Theoretical Properties of Adaptive Convolutions | Since Wf can take negative values, we will consider |Wf|^2 instead of Wf,
which is a probability density function, if properly normalized.
Let us start with approach (A).
Proposition 15
Let f\in W^{2,2}(\mathbb {R}^d,\mathbb {R})\setminus \lbrace 0\rbrace and Q\colon \mathbb {R}^d\rightarrow \mathrm {GL}(d,\mathbb {R... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.035733453929424286,
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0.014853338710963726,
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-0.004478125367313623,
0.01777518168091774,
... | |
055c74a78b2254487d768fa735f39b7bfae39171 | subsection | 19 | 108 | Theoretical Properties of Adaptive Convolutions | Remark 16 Here and in the following we will assume that the function f is such that
(REF ) has a unique symmetric and positive definite solution
\mu _f and that the corresponding fixed point iteration
\mu _f^{(n+1)}(x) = \sqrt{\frac{(\lambda \mu _f^{(n)})^2(x)}{2} + \frac{\left(
\nabla f\nabla f^{\intercal } - f\, D^... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.025749849155545235,
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0.03846450522542,
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-0.0031710322946310043,
-0.0041097491048276424,
0.00... | |
5e728ba960a8b1ef12f45a6627db5c46db15ad4d | subsection | 20 | 108 | Theoretical Properties of Adaptive Convolutions | Then, for each x\in \mathbb {R}^d, the
expectation value and covariance matrix of the probability distribution \mathbb {P}_{\rho _x} given
by the density
\rho _x(\xi ) = \frac{|Wf|^2(x,\xi )}{\Vert W f (x,{\cdot )\Vert _{L^2}^2}
are\mathrel {\mathop }
\mathbb {E}_{\rho _x} = 0
\qquad \text{and}\qquad \mathbb {C}\mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.017954502254724503,
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0.02613089419901371,
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0.022271515801548958,
0.005991190206259489,
... | |
796cbf74716d7ca63e6d2c26d9cb5abc1225bab7 | subsection | 21 | 108 | Theoretical Properties of Adaptive Convolutions | The scale invariance (Adaptation Axiom REF (A3)) of the choices (REF ) and (REF ) is clearly visible: f_2(x) = f_1(\alpha x) and, accordingly, \mu _f is \alpha =6 times higher in the `right' domain than in the `left' one.
[Figure: \mu _f^{(c)}, \mu _f^{(d)} and \mu _f^{(e)} as given by theformulas (), () and() describe... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
8ce621d4193d541bf21db3795aa4eedc6845b7e2 | subsection | 22 | 108 | Theoretical Properties of Adaptive Convolutions | \begin{}[H]
\centering \begin{}[b]{0.32}
\centering \includegraphics [width=]{images/three_gauss_density}
\caption {f from Example
\ref {example\mathrel {\mathop }threeGaussians}}
\end{}
\hfill \begin{}[b]{0.32}
\centering \includegraphics [width=]{images/three_gauss_adapted_convolution_FBI}
\caption {f\ast _{\mu _f} g... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.015084954909980297,
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... | |
85919a5b793e22c1054c69cd811f9e4371b6a6e9 | subsection | 23 | 108 | Theoretical Properties of Adaptive Convolutions | The choice (REF ) looked promising, but failed to capture
solely local properties of f and therefore, as demonstrated in Example
REF , could not realize a key property required for
adaptive convolutions:
If the function f = \sum _{k=1}^K f_k is the sum of several well-separated
functions f_1,\dots ,f_K, then its adapti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1805.01729",
"end": 1487,
"openalex_id": "https://openalex.org/W2802336905",
"raw": "I. Klebanov. Axiomatic approach to variable kernel density estimation. ArXiv e-prints, 2018.",
"source_ref_id": "5122feebb607cdd56ab37... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.013816805556416512,
-0.005355824250727892,
-0.03894035145640373,
-0.004520406946539879,
0.02287287823855877,
-0.006782517768442631,
-0.019607504829764366,
-0.02325434796512127,
0.020080525428056717,
0.02397150918841362,
-0.027114812284708023,
0.031951840966939926,
-0.015342681668698788,
... | |
4ab8d87a3bd30455706cc95baabee3d59dca8013 | subsection | 24 | 108 | Theoretical Properties of Adaptive Convolutions | \\
We define the \emph {h-adaptive convolutions of types two and three} by
\begin{align*}
(f\ast ^p[g\, |\, h])(x)
&=
f\bar{\ast }G_p\, ,
\qquad (f\ast ^p[g_1,g_2\, |\, h])(x)
=
f\bar{\ast }\tilde{G}_p \, ,
{where}
G_p(x,y)
&=
\left\Vert g\right\Vert _p
\frac{g\left(h(x)-h(y)\right)}{\left\Vert g(h({\cdot )-h(y))_p}\, ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.002643292536959052,
-0.0459543839097023,
0.0019128588028252125,
-0.0014885207638144493,
-0.017362579703330994,
-0.033138420432806015,
-0.0240451879799366,
0.01215227972716093,
0.03411487489938736,
0.013723760843276978,
-0.04326913505792618,
-0.007941320538520813,
0.0056107984855771065,
-... | |
747db171ce625ad1151cb710688a87c439a58994 | subsection | 25 | 108 | Theoretical Properties of Adaptive Convolutions | Note that \tilde{G} is symmetric, while G is not (see also Proposition
\ref {prop\mathrel {\mathop }symmetricG}). Both convolutions provide a strong smoothing close to
zero, and nearly no smoothing away from zero, where G and \tilde{G} act
nearly like Dirac \delta -distributions.}
\end{}
\right.
[Figure: Adaptive Conv... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03246845677495003,
-0.008765568025410175,
-0.015128043480217457,
-0.02412247844040394,
0.00629000086337328,
-0.036984749138355255,
-0.00910886749625206,
0.03472660109400749,
0.04681072756648064,
0.015250105410814285,
-0.030927427113056183,
-0.005927629768848419,
-0.004535361658781767,
-... | |
5cfa782b678eddb2a17568c088b04ce8fba60315 | subsection | 26 | 108 | Theoretical Properties of Adaptive Convolutions | \right.
\item If
g_1\in L^1\left(\mathbb {R}^d\times \mathbb {R}^n\right),\ g_2,\, g\in L^p\left(\mathbb {R}^d\times \mathbb {R}^n\right)
depend on an additional parameter in \mathbb {R}^d and
\begin{}
\item \displaystyle \int _{\mathbb {R}^d} |g_1(y,z-h(y))|\, \mathrm {d}y\le \Gamma _1 for
some constant \Gamma _1>0 (i... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.02982667274773121,
0.01246465090662241,
-0.00733461556956172,
-0.010839821770787239,
-0.00041073656757362187,
-0.003509020432829857,
0.0050384956412017345,
0.01815536618232727,
0.044671352952718735,
0.014264930970966816,
-0.053520187735557556,
-0.043664418160915375,
-0.005625875201076269,... | |
f07e5d21250561b1045178bee10b2a12fd9ccec3 | subsection | 27 | 108 | Theoretical Properties of Adaptive Convolutions | More precisely, in this case the
linearization of h at y, h(x)-h(y)\approx Dh(y)\, (x-y), is meaningful and yields
\begin{align*}
G_p(x,y)
&=
\left\Vert g\right\Vert _p
\frac{g\left(h(x)-h(y)\right)}{\left\Vert g(h({\cdot )-h(y))_p}
\approx \left\Vert g\right\Vert _p \frac{g\left(Dh(y)\, (x-y)\right)}{\left\Vert g(Dh(y... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01635691523551941,
-0.019698485732078552,
-0.006397048942744732,
-0.02656472846865654,
-0.008903227746486664,
-0.014289412647485733,
-0.02654946967959404,
0.00981872621923685,
0.04626321420073509,
-0.00469193235039711,
-0.029082350432872772,
-0.01888979598879814,
0.008964261040091515,
0... | |
3b56e996b6be944bbbac06c85fa50a7d285f56ce | subsection | 28 | 108 | Theoretical Properties of Adaptive Convolutions | For
example, we can `let a value f(y) contribute strongly to f\ast ^p[g\,
|\, h](x), even though x is far away from y (without contributing strongly to most values in
between)^{\prime } by choosing h such that h(y)\approx h(x), see Figure
\ref {fig\mathrel {\mathop }weighted_quadratic}.
}
\section {Proofs}
\right.\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01829780451953411,
0.003637813962996006,
0.006733866408467293,
-0.03394021466374397,
-0.021807808429002762,
-0.03610726073384285,
0.04379874840378761,
0.05423719435930252,
0.02537885680794716,
0.010179012082517147,
-0.05036092922091484,
0.014574147760868073,
-0.03772491589188576,
-0.006... | |
867936070c6e11dc1c3d71de6b9982ea409e4903 | subsection | 29 | 108 | Theoretical Properties of Adaptive Convolutions | Hölder^{\prime }s inequality yields
\begin{align*}
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{\frac{1}{p^{\prime }}}\, |f(y)|^{\frac{1}{p}}\,
|G(x,y)|\, \mathrm {d}y
\le \left\Vert |f|^{\frac{1}{p^{\prime }}}\right\Vert _{p^{\prime }}\, \left\Vert |f|^{\frac{1}{p}}\,
|G(x,{\cdot )|_{p},
}
which implies
\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.007306648883968592,
0.01929108053445816,
-0.019489485770463943,
-0.029241858050227165,
-0.019809985533356667,
-0.002937921555712819,
0.007165476214140654,
0.03708649054169655,
0.029714977368712425,
-0.014071499928832054,
-0.0776527002453804,
-0.0002345329412491992,
-0.028326142579317093,
... | |
01ca67089ddbcd66cc1b91253ca9613c51e1d14b | subsection | 30 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}\end{}}\right.}\end{align*}\begin{}[Proof of Corollary \ref {cor\mathrel {\mathop }young}]
First note that for p<\infty and y\in \mathbb {R}^d the change of variables formula
implies\mathrel {\mathop }
\begin{equation}
\left\Vert g_{\mu ,p}({\cdot ,y)_p^p
=
\int _{\mathbb {R}^d}|\det \mu (y)|\, \left| g\bi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.06435072422027588,
-0.005735243204981089,
-0.0025356414262205362,
-0.034434352070093155,
-0.03452593460679054,
-0.011737596243619919,
-0.012867092154920101,
0.044660866260528564,
0.005510107148438692,
-0.018835103139281273,
-0.0012134446296840906,
-0.01438580546528101,
-0.0032034174073487... | |
b02772d1008ea1e7be0b25dc5e73f5ea2a6485cb | subsection | 31 | 108 | Theoretical Properties of Adaptive Convolutions | Since
\frac{r-q}{qr} + \frac{r-p}{pr} + \frac{1}{r}
=
\frac{1}{q} - \frac{1}{r} + \frac{1}{p} - \frac{1}{r} + \frac{1}{r}
=
\frac{1}{q} + \frac{1}{p} - \frac{1}{r}
=
1,
the generalized Hölder's inequality yields
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{1-\frac{q}{r}}\, |f(y)|^{\frac{q}{r}}\,
|G(x,y)|^{... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.051741477102041245,
0.021093428134918213,
-0.019597657024860382,
-0.01784241572022438,
0.0030297001358121634,
0.0026710203383117914,
-0.0007268990739248693,
0.02225341461598873,
0.01205774862319231,
-0.006643209140747786,
-0.055099330842494965,
0.004208764992654324,
-0.025244956836104393,... | |
8bd03e71ceefddb15d7975d0e35a8e5271453f4e | subsection | 32 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}
\right.
}\right.}
[Proof of Proposition REF ]
For all j=1,\dots ,d and \alpha \in \mathbb {N}^d with |\alpha |<m, we have by
induction:
\partial _{x_j}\partial ^\alpha \left(f\ast _{\mu }^p g\right)(x)
&=
\partial _{x_j}\left(\int _{\mathbb {R}^d} f(y)\, |\det (\mu (y))|^{1/p}\,
\alpha (\mu (y))\, D^{|\al... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.00034682522527873516,
0.007345543708652258,
-0.03786425665020943,
-0.026697199791669846,
-0.013668966479599476,
0.009328764863312244,
0.013821521773934364,
0.006102217361330986,
0.0364302359521389,
0.0036250983830541372,
-0.018474461510777473,
-0.0026906963903456926,
-0.006914574652910232... | |
1659ad4cab40334ef69029d3d84642137e106957 | subsection | 33 | 108 | Theoretical Properties of Adaptive Convolutions | The observations
\begin{}
\begin{align*}
\left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d j_t(y)
&=
\sum _{k=1}^d {\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]\, j_{k,t}(y)
=
{\rm tr}\left[\mu _t^{-1}(y)\sum _{k=1}^d\partial _{k}\mu _t(y)\, j_{k,t}(y)\right]
\\
&=
{\rm tr}\lef... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.00036994070978835225,
0.028954533860087395,
-0.03228018805384636,
-0.01919114962220192,
-0.014095884747803211,
0.010213414207100868,
0.02962576597929001,
0.02478983998298645,
0.02382875792682171,
0.046559132635593414,
-0.02880198135972023,
0.003932049963623285,
-0.003943491727113724,
-0.... | |
11df493623466f681ce658e487f31466d1db8315 | subsection | 34 | 108 | Theoretical Properties of Adaptive Convolutions | \end{pmatrix}
lead to
\begin{align*}
&\left(\nabla _y\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\right)^{\intercal }j_t(y)
\\
&=
\delta _t(y) \left(
g_{\mu _t}(x,y) \left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d + (\nabla g)_{\mu _t}(x,y)^{\intercal }
\left[M_t(x,y)-\mu _t(y)\right] \ri... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.0151362968608737,
0.012786508537828922,
-0.005977463908493519,
-0.03228669613599777,
-0.017928577959537506,
0.038786761462688446,
0.013343438506126404,
0.02511526644229889,
0.0034064296633005142,
0.03222566470503807,
-0.008926142938435078,
0.0026072729378938675,
-0.008956659585237503,
0... | |
9295bcfb524f51391c7b383959b0898b91e0e915 | subsection | 35 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}
Combining these two, we get\mathrel {\mathop }
\begin{align*}
\partial _t \rho _{g,t}(x)
&=
\int _{\mathbb {R}^d} \underbrace{\partial _t\rho _t(y)}_{= -\operatorname{div}j_t(y)}
\delta _t(y) \, g_{\mu _t}(x,y)
+ \partial _t\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\, \rho _t(y)
\, \mathrm {d}y
\\
&=
\int... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.015744362026453018,
0.011442123912274837,
0.0023551704362034798,
-0.017559845000505447,
-0.002164468402042985,
-0.008520567789673805,
0.00038569490425288677,
0.02915453165769577,
-0.0009239514474757016,
0.01267024502158165,
-0.002175910398364067,
-0.010175861418247223,
-0.0237996168434619... | |
e7752c9f2c69e225395ebdd893d7a914faee24db | subsection | 36 | 108 | Theoretical Properties of Adaptive Convolutions | We have:
&(f({\cdot \, - a)\ast ^p_{\mu _{f({\cdot \, - a)}} g) (x)
=
\int f(y-a) \left|\det \left(\mu _{f}(y-a)\right)\right|
g\left(\mu _{f}(y-a)(x-y)\right)\, \mathrm {d}y
\\
& \hspace{28.45274pt} =
\int f(y) \left|\det \left(\mu _{f}(y)\right)\right|
g\left(\mu _{f}(y)(x-a-y)\right)\, \mathrm {d}y
=
(f\ast ^p_{\mu... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005618390627205372,
0.003526274347677827,
-0.021588657051324844,
-0.05312487855553627,
-0.010733299888670444,
0.009733966551721096,
-0.01843045838177204,
0.019681531935930252,
0.02998000755906105,
0.006201970856636763,
-0.030468231067061424,
0.03019360452890396,
-0.010618872940540314,
0... | |
f543aa65db30f3f79b4dfa93eced4dffd99ff256 | subsection | 37 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}
\right.
}\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFourier}]
The proof is analogous to the one of Proposition
\ref {prop\mathrel {\mathop }muWigner2}.
\end{}
\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFBI}]
The proof is analogous to the one of Proposition
... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.044177960604429245,
-0.00014349065895657986,
-0.025017905980348587,
-0.003083380637690425,
-0.023324621841311455,
-0.021188946440815926,
0.019160054624080658,
0.03499456122517586,
0.06748732924461365,
0.04579497128725052,
-0.03325551003217697,
-0.012043680995702744,
0.00414549745619297,
... | |
0f365209950676fb3cfac99d4950c697da3f484c | subsection | 38 | 108 | Theoretical Properties of Adaptive Convolutions | Then we have for x,y\in \mathbb {R}^d
\big (\nabla \tilde{f}\nabla \tilde{f}^{\intercal } - \tilde{f}\, D^2 \tilde{f}\big )(x)
=
A^\intercal \, \big (\nabla f\nabla f^{\intercal } - f\, D^2 f\big )(Ax)\, A,
\qquad G_{\mu _{\tilde{f}}^{-2}(x)}(y)
=
*{\det A}\, G_{\mu _f^{-2}(Ax)}(Ay),
and, since for any functions \phi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.014687075279653072,
0.009872766211628914,
0.009514173492789268,
-0.0009026829502545297,
-0.006878129206597805,
-0.002988915191963315,
0.008552837185561657,
-0.010902768932282925,
0.07165767252445221,
0.024826880544424057,
-0.033021122217178345,
-0.024414878338575363,
0.0002694220165722072,... | |
d206ba9a1179f775c00bb8a31719c5bd9ad4728b | subsection | 39 | 108 | Theoretical Properties of Adaptive Convolutions | Therefore,
\mu _{f^{(t)}}(x+a_k^{(t)})
\stackrel{(A1)}{=}
\mu _{\tilde{f}^{(t)}}(x)
\xrightarrow{}
\mu _{f_k}(x).
}
}}}\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muWigner2}]
As the Wigner transform is real-valued, we get for real-valued functions~f\mathrel {\mathop }
\begin{align*}
W f(x,-\xi )
&=
(2\... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005172934383153915,
-0.004856301937252283,
-0.011620028875768185,
-0.014893168583512306,
-0.01591554842889309,
0.0057489764876663685,
0.015656137838959694,
0.03415052220225334,
0.046693746000528336,
0.026490308344364166,
0.0038415524177253246,
0.014999983832240105,
0.0035401794593781233,
... | |
ba862faf0cf197c69105f1aae56f04951a136844 | subsection | 40 | 108 | Theoretical Properties of Adaptive Convolutions | For the covariance matrix, we use the transformation
z_1 = y_1-y_2,\qquad z_2 = y_1+y_2
and the function
F(z_1,z_2) = f\left(x+\frac{z_2 + z_1}{4}\right)f\left(x-\frac{z_2 +
z_1}{4}\right)f\left(x+\frac{z_2 - z_1}{4}\right)f\left(x-\frac{z_2 -
z_1}{4}\right)
to compute\mathrel {\mathop }
\begin{}
\begin{align*}
\in... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005715889856219292,
-0.0007206866866908967,
0.0014432811876758933,
0.03531793877482414,
0.008554755710065365,
0.0049985419027507305,
0.03183804079890251,
0.022527778521180153,
-0.011828609742224216,
0.023825110867619514,
-0.049848053604364395,
-0.01061522401869297,
0.002384800463914871,
... | |
679f3941f02d2c916c5bc9ae54645c70d42e0e14 | subsection | 41 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}
\end{}
Taking the quotient proves the formula for the covariance matrix.
\end{}
\right.}\end{}
[Proof of Theorem REF ]
Adaptation Axiom REF (A1) follows from
\left(f({\cdot -a)\ast g({\cdot -b)(x)
}}\right.&=
\int f(y-a)\, g(x-y-b)\, \mathrm {d}y
=
\int f(y)\, g(x-(a+b)-y)\, \mathrm {d}y
\\
&=
(f\ast g)(x... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.0013511285651475191,
-0.00341167114675045,
-0.029612313956022263,
-0.011846451088786125,
0.0258440300822258,
0.0018088150536641479,
0.03987974673509598,
0.038262587040662766,
0.020611146464943886,
-0.006117742508649826,
-0.03551647067070007,
0.0018316993955522776,
-0.021862156689167023,
... | |
25260436030c284b1de9a9175d43aaa4c8245f01 | subsection | 42 | 108 | Theoretical Properties of Adaptive Convolutions | The claim follows from the definitions (REF ) of
\mu _f^{(d)} and (REF ) of
\mu _f^{(e)}.
[Proof of Proposition ]
For the h-adaptive convolution of type two the property \left\Vert
G({\cdot ,y)_p = \left\Vert g\right\Vert _p is straightforward for all
y\in \mathbb {R}^d, 1\le p \le \infty and Theorem \ref {theorem\mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03579450771212578,
0.008384092710912228,
-0.010909237898886204,
-0.02456485852599144,
-0.020018070936203003,
-0.027677424252033234,
0.006740078330039978,
0.03478750213980675,
0.03912068158388138,
0.005679669789969921,
-0.04543735831975937,
-0.005008331965655088,
-0.0037705532740801573,
... | |
3f0e72e28181ae36ca109d335c88a6f1e1bc7b6c | subsection | 43 | 108 | Theoretical Properties of Adaptive Convolutions | Hölder^{\prime }s inequality yields
\begin{align*}
|\tilde{G}(x,y)|
&\le \left\Vert g_2\right\Vert _p \int |g_1(z-h(y))|^{1/p^{\prime }}\, |g_1(z-h(y))|^{1/p}
\gamma (x,z)\, \mathrm {d}z
\\
&\le \left\Vert g_2\right\Vert _p \left\Vert g_1({\cdot -h(y))^{1/p^{\prime }}_{p^{\prime }}
\left\Vert g_1({\cdot -h(y))^{1/p}\, ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.023305442184209824,
0.013987843878567219,
-0.02129082754254341,
-0.02271021530032158,
-0.027670443058013916,
0.0026613534428179264,
0.020527714863419533,
0.03063131868839264,
0.01092776469886303,
-0.008707108907401562,
-0.049419138580560684,
0.00428105890750885,
-0.03565259650349617,
-0... | |
d97e3a295a4c8f983f2e54a97136cf7e401e3ee0 | subsection | 44 | 108 | Theoretical Properties of Adaptive Convolutions | \end{align*}
\end{}
Therefore \left\Vert \tilde{G}({\cdot ,y)_p \le \left\Vert g_1\right\Vert _1
\left\Vert g_2\right\Vert _p (for all y\in \mathbb {R}^d and 1\le p\le \infty ) also holds for
type three and again Theorem \ref {theorem\mathrel {\mathop }young1} proves the claim.
}
\right.}\right.}\right.\begin{}[Proof o... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.05638016015291214,
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-0.0007612885092385113,
-0.008588211610913277,
... | |
e278445808f8181aefd6988d8b298bf828132384 | subsection | 45 | 108 | Theoretical Properties of Adaptive Convolutions | Springer, 1997.
R. A. Horn and C. R. Johnson.
Matrix analysis.
Cambridge University Press, 2012.
I. Klebanov.
Axiomatic approach to variable kernel density estimation.
ArXiv e-prints, 2018.
W. Young.
On the multiplication of successions of fourier constants.
Proceedings of the Royal Society of London. Series A, Contai... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.01570102758705616,
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-0.00283236475661397,
-0.022399522364139557,
... | |
dbd3b35758caca56ca5c6d04dbaabe0c4f36fcaa | subsection | 46 | 108 | Automatic Choice of the Adaptation Function | The adaptation function \mu in Example REF was
chosen manually for the adaptive smoothing of a function f\in L^1(\mathbb {R}^d).
Let us now discuss how this choice can be performed automatically in
dependence of the function f that we want to smooth. To this end, we will have
to get a grip on the local variation of f, ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1396,
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"raw": "R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, 2012.",
"source_ref_id": "6e1fd94cd82860a43a9f3434762507741c7ba0cf",
"start": ... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.011862807907164097,
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0.008483242243528366,
-0.011679716408252716,
... | |
daad684aecae2490b61b91595845f542d3acc317 | subsection | 47 | 108 | Automatic Choice of the Adaptation Function | We say that a mapping\mathtt {m}\colon W^{2,2}(\mathbb {R}^d,\mathbb {R})\rightarrow \mathcal {M},
\qquad f\mapsto \mu _f,fulfills the Adaptation Axioms, if for any a\in \mathbb {R}^d,
\alpha \in \mathbb {R}\setminus \lbrace 0\rbrace , A\in \mathrm {GL}(d,\mathbb {R}), any parametrized function
f^{(t)} = \sum _{k=1}^K ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.010439352132380009,
-0.0031230300664901733,
-0.029990244656801224,
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0.0217028651386499... | |
c354b2425a1df19883ea334b04017fe7c75b9546 | subsection | 48 | 108 | Automatic Choice of the Adaptation Function | These consequences are stated in the following theorem\mathrel {\mathop }
\begin{}
Assuming Adaptation Axioms \ref {cond\mathrel {\mathop }adaptation} (A1)--(A4) and adopting that
notation, we have for any f\in W^{2,2}(\mathbb {R}^d,\mathbb {R}), radially symmetric g\in L^p and x\in \mathbb {R}^d\mathrel {\mathop }
\b... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.02234596572816372,
0.03309156745672226,
-0.029779357835650444,
-0.019339028745889664,
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0.006712184753268957,
-0.031687311828136444,
-0... | |
1f3703fb0454177e42654655e81c0c285a56700b | subsection | 49 | 108 | Automatic Choice of the Adaptation Function | However, \nabla f\nabla f^{\intercal } is only positive \emph {semi}-definite
and therefore \mu _f^{(a)}\in \mathrm {GL}(d,\mathbb {R}) could be violated. Also, it
is unclear in how far the Adaptation Axiom (A5) is fulfilled.
Obviously, this last axiom is not rigorous and we will discuss it now. In order
to get a grasp... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.004669299349188805,
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... | |
afc4a7dcf0a516fa8e980389b69e2c498f3aaf7a | subsection | 50 | 108 | Phase Space Transformations | In this section, we will discuss four transformations, which will allow us to quantify the (local) variation of a function.
All four transformations can be viewed from various perspectives and we will
focus on the time-frequency point of view. In the following, \mathcal {S}(\mathbb {R}^d,\mathbb {C}) will denote the Sc... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04541068524122238,
0.04019211605191231,
-0.026474306359887123,
-0.013275301083922386,
0.003450433723628521,
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0.061157938092947006,
0.016052432358264923,
-0.005550449248403311,
0.0035553390625864267,
... | |
c08c38c738161de929c9a1e358226505209a2cb4 | subsection | 51 | 108 | Phase Space Transformations | G[a,\Sigma ](x)
&=
(2\pi )^{-d/2}\, |\det \Sigma |^{-1/2}\,
\exp \left[-\frac{1}{2}(x-a)^{-\intercal }\Sigma ^{-1} (x-a)\right],
\\
\mathcal {F} f (\xi )
&=
(2\pi )^{-d/2} \int _{\mathbb {R}^d} f(y)\, e^{-iy^\intercal \xi }\,
\mathrm {d}y\, ,
\\
\mathcal {F}_{Q} f(x,\xi )
&=
\pi ^{-d/4} \int _{\mathbb {R}^d}f(y)\, G_{... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04087414592504501,
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-0.013042178936302662,
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0.028389062732458115,
0.017430273815989494,
0.01... | |
42a1751d7190eb89363238ca9ad76aa5c2b2e472 | subsection | 52 | 108 | Phase Space Transformations | The technique of using (2\pi )^{-d}\int _{\mathbb {R}^d} e^{-ix^\intercal \xi }\, \mathrm {d}\xi as a \delta -distribution will be used in the proofs of Propositions
REF , REF , REF and
REF .
From the point of view of time-frequency analysis, the Fourier transform yields a
decomposition of the signal f into its frequen... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04631495103240013,
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0.0679163932800293,
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0.032158076763153076,
0.012585584074258804,
0.028... | |
c68d603e57905c592de330aa972783b83aa7f715 | subsection | 53 | 108 | Phase Space Transformations | Then the Wigner transform Wf of f fulfills
\begin{align*}
&\int _{\mathbb {R}^d} Wf(x,\xi )\, \mathrm {d}\xi \ \ =\ |f(x)|^2\, ,
\qquad \int _{\mathbb {R}^d} Wf(x,\xi )\, \mathrm {d}x
\ =\ |\mathcal {F}f(\xi )|^2\, ,
{and}
&Wf \ast G_{\sqrt{1/2}} = |\mathcal {\mathcal {F}}_{1} f|^2.
\end{align*}
\end{}
See .
Transform... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.0012876334367319942,
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-0.038457319140434265,
0.021227829158306122,
-0.006852117832750082,
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-0.011621130630373955,
... | |
86d4f58556542394d1185842f0fa1ec123d6415d | subsection | 54 | 108 | Phase Space Transformations | The expectation value and covariance
matrix of the probability distribution \mathbb {P}_\rho given by the spectral density
\rho = \frac{|\mathcal {F}f|^2}{\Vert \mathcal {F}f\Vert _{L^2}^2} = \frac{|\mathcal {F}f|^2}{\Vert f\Vert _{L^2}^2}
(here we used the Plancherel theorem REF ) are:
\mathbb {E}_{\rho } = 0
\qqua... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.006337917875498533,
-0.02240593172609806,
-0.013797658495604992,
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0.03125841170549393,
0.0006129007088020444,
... | |
33f026dc85da5f157ba0dbabbb3778e2f0a0030e | subsection | 55 | 108 | Phase Space Transformations | Then, for each x\in \mathbb {R}^d, the expectation value and covariance matrix of the
probability distribution \mathbb {P}_{\rho _x} given by the density
\rho _x(\xi ) = \frac{|\mathcal {F}_Q f(x,\xi )|^2}{\Vert \mathcal {F}_Q f (x,{\cdot )\Vert _{L^2}^2}
are\mathrel {\mathop }
\mathbb {E}_{\rho _x} = 0
\qquad \text... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03940245509147644,
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... | |
5a36906068cff00f44b9e37edd3516d55655a4ea | subsection | 56 | 108 | Phase Space Transformations | Proposition 15
Let f\in W^{2,2}(\mathbb {R}^d,\mathbb {R})\setminus \lbrace 0\rbrace and Q\colon \mathbb {R}^d\rightarrow \mathrm {GL}(d,\mathbb {R}). Then,
for each x\in \mathbb {R}^d, the expectation value and covariance matrix of the probability distribution \mathbb {P}_{\rho _x} given
by the density
\rho _x(\xi )... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03413942828774452,
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0.011524725705385208,
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0.... | |
c3fccdb88beaa0f36492c10f3d99bdf3532abe3d | subsection | 57 | 108 | Phase Space Transformations | Remark 16 Here and in the following we will assume that the function f is such that
(REF ) has a unique symmetric and positive definite solution
\mu _f and that the corresponding fixed point iteration
\mu _f^{(n+1)}(x) = \sqrt{\frac{(\lambda \mu _f^{(n)})^2(x)}{2} + \frac{\left(
\nabla f\nabla f^{\intercal } - f\, D^... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.025749849155545235,
0.01944594457745552,
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0.00... | |
bae25ae7217a370f78da6f557ed54318ce231ac6 | subsection | 58 | 108 | Phase Space Transformations | Then, for each x\in \mathbb {R}^d, the
expectation value and covariance matrix of the probability distribution \mathbb {P}_{\rho _x} given
by the density
\rho _x(\xi ) = \frac{|Wf|^2(x,\xi )}{\Vert W f (x,{\cdot )\Vert _{L^2}^2}
are\mathrel {\mathop }
\mathbb {E}_{\rho _x} = 0
\qquad \text{and}\qquad \mathbb {C}\mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.017954502254724503,
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0.022271515801548958,
0.005991190206259489,
... | |
f9da81e7d138eb139058cf8cfac111f6fdacdc61 | subsection | 59 | 108 | Phase Space Transformations | The scale invariance (Adaptation Axiom REF (A3)) of the choices (REF ) and (REF ) is clearly visible: f_2(x) = f_1(\alpha x) and, accordingly, \mu _f is \alpha =6 times higher in the `right' domain than in the `left' one.
[Figure: \mu _f^{(c)}, \mu _f^{(d)} and \mu _f^{(e)} as given by theformulas (), () and() describe... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01773620955646038,
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... | |
a3aa14a74af4339d6210114ab3111f0fa1a84dac | subsection | 60 | 108 | Phase Space Transformations | \begin{}[H]
\centering \begin{}[b]{0.32}
\centering \includegraphics [width=]{images/three_gauss_density}
\caption {f from Example
\ref {example\mathrel {\mathop }threeGaussians}}
\end{}
\hfill \begin{}[b]{0.32}
\centering \includegraphics [width=]{images/three_gauss_adapted_convolution_FBI}
\caption {f\ast _{\mu _f} g... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.015084954909980297,
-0.009392807260155678,
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... | |
5b5b370d772524310e6a15dde88690cd0e6c25e6 | subsection | 61 | 108 | Phase Space Transformations | The choice (REF ) looked promising, but failed to capture
solely local properties of f and therefore, as demonstrated in Example
REF , could not realize a key property required for
adaptive convolutions:
If the function f = \sum _{k=1}^K f_k is the sum of several well-separated
functions f_1,\dots ,f_K, then its adapti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1805.01729",
"end": 1487,
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"raw": "I. Klebanov. Axiomatic approach to variable kernel density estimation. ArXiv e-prints, 2018.",
"source_ref_id": "5122feebb607cdd56ab37... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.013816805556416512,
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-0.015342681668698788,
... | |
cc754f8c2ee4693c5e589250760c1fd84b3c8748 | subsection | 62 | 108 | Phase Space Transformations | \\
We define the \emph {h-adaptive convolutions of types two and three} by
\begin{align*}
(f\ast ^p[g\, |\, h])(x)
&=
f\bar{\ast }G_p\, ,
\qquad (f\ast ^p[g_1,g_2\, |\, h])(x)
=
f\bar{\ast }\tilde{G}_p \, ,
{where}
G_p(x,y)
&=
\left\Vert g\right\Vert _p
\frac{g\left(h(x)-h(y)\right)}{\left\Vert g(h({\cdot )-h(y))_p}\, ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.013723760843276978,
-0.04326913505792618,
-0.007941320538520813,
0.0056107984855771065,
-... | |
7ec6fccd771536b2081670da16b6184b5cf3e718 | subsection | 63 | 108 | Phase Space Transformations | Note that \tilde{G} is symmetric, while G is not (see also Proposition
\ref {prop\mathrel {\mathop }symmetricG}). Both convolutions provide a strong smoothing close to
zero, and nearly no smoothing away from zero, where G and \tilde{G} act
nearly like Dirac \delta -distributions.}
\end{}
\right.
[Figure: Adaptive Conv... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03246845677495003,
-0.008765568025410175,
-0.015128043480217457,
-0.02412247844040394,
0.00629000086337328,
-0.036984749138355255,
-0.00910886749625206,
0.03472660109400749,
0.04681072756648064,
0.015250105410814285,
-0.030927427113056183,
-0.005927629768848419,
-0.004535361658781767,
-... | |
dc6a495c7a8b4285a329c903dff2453b6816b33f | subsection | 64 | 108 | Phase Space Transformations | \right.
\item If
g_1\in L^1\left(\mathbb {R}^d\times \mathbb {R}^n\right),\ g_2,\, g\in L^p\left(\mathbb {R}^d\times \mathbb {R}^n\right)
depend on an additional parameter in \mathbb {R}^d and
\begin{}
\item \displaystyle \int _{\mathbb {R}^d} |g_1(y,z-h(y))|\, \mathrm {d}y\le \Gamma _1 for
some constant \Gamma _1>0 (i... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.02982667274773121,
0.01246465090662241,
-0.00733461556956172,
-0.010839821770787239,
-0.00041073656757362187,
-0.003509020432829857,
0.0050384956412017345,
0.01815536618232727,
0.044671352952718735,
0.014264930970966816,
-0.053520187735557556,
-0.043664418160915375,
-0.005625875201076269,... | |
e83d6af0bf0dc66793202b0f32c5587ec2ae3f78 | subsection | 65 | 108 | Phase Space Transformations | More precisely, in this case the
linearization of h at y, h(x)-h(y)\approx Dh(y)\, (x-y), is meaningful and yields
\begin{align*}
G_p(x,y)
&=
\left\Vert g\right\Vert _p
\frac{g\left(h(x)-h(y)\right)}{\left\Vert g(h({\cdot )-h(y))_p}
\approx \left\Vert g\right\Vert _p \frac{g\left(Dh(y)\, (x-y)\right)}{\left\Vert g(Dh(y... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01635691523551941,
-0.019698485732078552,
-0.006397048942744732,
-0.02656472846865654,
-0.008903227746486664,
-0.014289412647485733,
-0.02654946967959404,
0.00981872621923685,
0.04626321420073509,
-0.00469193235039711,
-0.029082350432872772,
-0.01888979598879814,
0.008964261040091515,
0... | |
5b0cf313f509d904f12db56c7e9e5dc2274a26d4 | subsection | 66 | 108 | Phase Space Transformations | For
example, we can `let a value f(y) contribute strongly to f\ast ^p[g\,
|\, h](x), even though x is far away from y (without contributing strongly to most values in
between)^{\prime } by choosing h such that h(y)\approx h(x), see Figure
\ref {fig\mathrel {\mathop }weighted_quadratic}.
}
\section {Proofs}
\right.\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01829780451953411,
0.003637813962996006,
0.006733866408467293,
-0.03394021466374397,
-0.021807808429002762,
-0.03610726073384285,
0.04379874840378761,
0.05423719435930252,
0.02537885680794716,
0.010179012082517147,
-0.05036092922091484,
0.014574147760868073,
-0.03772491589188576,
-0.006... | |
56ba03e244ec9129b435a2be923ab774e7b73fda | subsection | 67 | 108 | Phase Space Transformations | Hölder^{\prime }s inequality yields
\begin{align*}
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{\frac{1}{p^{\prime }}}\, |f(y)|^{\frac{1}{p}}\,
|G(x,y)|\, \mathrm {d}y
\le \left\Vert |f|^{\frac{1}{p^{\prime }}}\right\Vert _{p^{\prime }}\, \left\Vert |f|^{\frac{1}{p}}\,
|G(x,{\cdot )|_{p},
}
which implies
\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.007306648883968592,
0.01929108053445816,
-0.019489485770463943,
-0.029241858050227165,
-0.019809985533356667,
-0.002937921555712819,
0.007165476214140654,
0.03708649054169655,
0.029714977368712425,
-0.014071499928832054,
-0.0776527002453804,
-0.0002345329412491992,
-0.028326142579317093,
... | |
b7b9fd89ccb7f5d4cea8dda4725b2285ae2244c8 | subsection | 68 | 108 | Phase Space Transformations | \end{align*}\end{}}\right.}\end{align*}\begin{}[Proof of Corollary \ref {cor\mathrel {\mathop }young}]
First note that for p<\infty and y\in \mathbb {R}^d the change of variables formula
implies\mathrel {\mathop }
\begin{equation}
\left\Vert g_{\mu ,p}({\cdot ,y)_p^p
=
\int _{\mathbb {R}^d}|\det \mu (y)|\, \left| g\bi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.06435072422027588,
-0.005735243204981089,
-0.0025356414262205362,
-0.034434352070093155,
-0.03452593460679054,
-0.011737596243619919,
-0.012867092154920101,
0.044660866260528564,
0.005510107148438692,
-0.018835103139281273,
-0.0012134446296840906,
-0.01438580546528101,
-0.0032034174073487... | |
479d2fbf1b32f9f44add844a142dba0ab28d7031 | subsection | 69 | 108 | Phase Space Transformations | Since
\frac{r-q}{qr} + \frac{r-p}{pr} + \frac{1}{r}
=
\frac{1}{q} - \frac{1}{r} + \frac{1}{p} - \frac{1}{r} + \frac{1}{r}
=
\frac{1}{q} + \frac{1}{p} - \frac{1}{r}
=
1,
the generalized Hölder's inequality yields
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{1-\frac{q}{r}}\, |f(y)|^{\frac{q}{r}}\,
|G(x,y)|^{... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.051741477102041245,
0.021093428134918213,
-0.019597657024860382,
-0.01784241572022438,
0.0030297001358121634,
0.0026710203383117914,
-0.0007268990739248693,
0.02225341461598873,
0.01205774862319231,
-0.006643209140747786,
-0.055099330842494965,
0.004208764992654324,
-0.025244956836104393,... | |
bf452d313475cc2adcc765ebf14d953818a1dbcf | subsection | 70 | 108 | Phase Space Transformations | \end{align*}
\right.
}\right.}
[Proof of Proposition REF ]
For all j=1,\dots ,d and \alpha \in \mathbb {N}^d with |\alpha |<m, we have by
induction:
\partial _{x_j}\partial ^\alpha \left(f\ast _{\mu }^p g\right)(x)
&=
\partial _{x_j}\left(\int _{\mathbb {R}^d} f(y)\, |\det (\mu (y))|^{1/p}\,
\alpha (\mu (y))\, D^{|\al... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.00034682522527873516,
0.007345543708652258,
-0.03786425665020943,
-0.026697199791669846,
-0.013668966479599476,
0.009328764863312244,
0.013821521773934364,
0.006102217361330986,
0.0364302359521389,
0.0036250983830541372,
-0.018474461510777473,
-0.0026906963903456926,
-0.006914574652910232... | |
064e75a7faa16587af353f8b74af0d237ea74144 | subsection | 71 | 108 | Phase Space Transformations | The observations
\begin{}
\begin{align*}
\left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d j_t(y)
&=
\sum _{k=1}^d {\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]\, j_{k,t}(y)
=
{\rm tr}\left[\mu _t^{-1}(y)\sum _{k=1}^d\partial _{k}\mu _t(y)\, j_{k,t}(y)\right]
\\
&=
{\rm tr}\lef... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.00036994070978835225,
0.028954533860087395,
-0.03228018805384636,
-0.01919114962220192,
-0.014095884747803211,
0.010213414207100868,
0.02962576597929001,
0.02478983998298645,
0.02382875792682171,
0.046559132635593414,
-0.02880198135972023,
0.003932049963623285,
-0.003943491727113724,
-0.... | |
28cd42a9059ed8909bdf410dd2dff65b5a1641b2 | subsection | 72 | 108 | Phase Space Transformations | \end{pmatrix}
lead to
\begin{align*}
&\left(\nabla _y\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\right)^{\intercal }j_t(y)
\\
&=
\delta _t(y) \left(
g_{\mu _t}(x,y) \left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d + (\nabla g)_{\mu _t}(x,y)^{\intercal }
\left[M_t(x,y)-\mu _t(y)\right] \ri... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.0151362968608737,
0.012786508537828922,
-0.005977463908493519,
-0.03228669613599777,
-0.017928577959537506,
0.038786761462688446,
0.013343438506126404,
0.02511526644229889,
0.0034064296633005142,
0.03222566470503807,
-0.008926142938435078,
0.0026072729378938675,
-0.008956659585237503,
0... | |
b220ec86e2b42cbe9e557d5b242fb4335cf07f41 | subsection | 73 | 108 | Phase Space Transformations | \end{align*}
Combining these two, we get\mathrel {\mathop }
\begin{align*}
\partial _t \rho _{g,t}(x)
&=
\int _{\mathbb {R}^d} \underbrace{\partial _t\rho _t(y)}_{= -\operatorname{div}j_t(y)}
\delta _t(y) \, g_{\mu _t}(x,y)
+ \partial _t\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\, \rho _t(y)
\, \mathrm {d}y
\\
&=
\int... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.015744362026453018,
0.011442123912274837,
0.0023551704362034798,
-0.017559845000505447,
-0.002164468402042985,
-0.008520567789673805,
0.00038569490425288677,
0.02915453165769577,
-0.0009239514474757016,
0.01267024502158165,
-0.002175910398364067,
-0.010175861418247223,
-0.0237996168434619... | |
d9d8782c429b0ec7f25ef3d25ac6be601b8b344b | subsection | 74 | 108 | Phase Space Transformations | We have:
&(f({\cdot \, - a)\ast ^p_{\mu _{f({\cdot \, - a)}} g) (x)
=
\int f(y-a) \left|\det \left(\mu _{f}(y-a)\right)\right|
g\left(\mu _{f}(y-a)(x-y)\right)\, \mathrm {d}y
\\
& \hspace{28.45274pt} =
\int f(y) \left|\det \left(\mu _{f}(y)\right)\right|
g\left(\mu _{f}(y)(x-a-y)\right)\, \mathrm {d}y
=
(f\ast ^p_{\mu... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005618390627205372,
0.003526274347677827,
-0.021588657051324844,
-0.05312487855553627,
-0.010733299888670444,
0.009733966551721096,
-0.01843045838177204,
0.019681531935930252,
0.02998000755906105,
0.006201970856636763,
-0.030468231067061424,
0.03019360452890396,
-0.010618872940540314,
0... | |
3a76262ad9081d6aa0cf4c519a9c7e10bd33bbd5 | subsection | 75 | 108 | Phase Space Transformations | \end{align*}
\right.
}\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFourier}]
The proof is analogous to the one of Proposition
\ref {prop\mathrel {\mathop }muWigner2}.
\end{}
\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFBI}]
The proof is analogous to the one of Proposition
... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.044177960604429245,
-0.00014349065895657986,
-0.025017905980348587,
-0.003083380637690425,
-0.023324621841311455,
-0.021188946440815926,
0.019160054624080658,
0.03499456122517586,
0.06748732924461365,
0.04579497128725052,
-0.03325551003217697,
-0.012043680995702744,
0.00414549745619297,
... | |
37d2d0d4bdab247fb4b8d43cf82e6a161e4e2a2a | subsection | 76 | 108 | Phase Space Transformations | Then we have for x,y\in \mathbb {R}^d
\big (\nabla \tilde{f}\nabla \tilde{f}^{\intercal } - \tilde{f}\, D^2 \tilde{f}\big )(x)
=
A^\intercal \, \big (\nabla f\nabla f^{\intercal } - f\, D^2 f\big )(Ax)\, A,
\qquad G_{\mu _{\tilde{f}}^{-2}(x)}(y)
=
*{\det A}\, G_{\mu _f^{-2}(Ax)}(Ay),
and, since for any functions \phi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.014687075279653072,
0.009872766211628914,
0.009514173492789268,
-0.0009026829502545297,
-0.006878129206597805,
-0.002988915191963315,
0.008552837185561657,
-0.010902768932282925,
0.07165767252445221,
0.024826880544424057,
-0.033021122217178345,
-0.024414878338575363,
0.0002694220165722072,... | |
5dead821fe21e88897ee2370059ba3541e9d0a92 | subsection | 77 | 108 | Phase Space Transformations | Therefore,
\mu _{f^{(t)}}(x+a_k^{(t)})
\stackrel{(A1)}{=}
\mu _{\tilde{f}^{(t)}}(x)
\xrightarrow{}
\mu _{f_k}(x).
}
}}}\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muWigner2}]
As the Wigner transform is real-valued, we get for real-valued functions~f\mathrel {\mathop }
\begin{align*}
W f(x,-\xi )
&=
(2\... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005172934383153915,
-0.004856301937252283,
-0.011620028875768185,
-0.014893168583512306,
-0.01591554842889309,
0.0057489764876663685,
0.015656137838959694,
0.03415052220225334,
0.046693746000528336,
0.026490308344364166,
0.0038415524177253246,
0.014999983832240105,
0.0035401794593781233,
... | |
1cd6f6ab6bac4b11e29d85b791911988915afc55 | subsection | 78 | 108 | Phase Space Transformations | For the covariance matrix, we use the transformation
z_1 = y_1-y_2,\qquad z_2 = y_1+y_2
and the function
F(z_1,z_2) = f\left(x+\frac{z_2 + z_1}{4}\right)f\left(x-\frac{z_2 +
z_1}{4}\right)f\left(x+\frac{z_2 - z_1}{4}\right)f\left(x-\frac{z_2 -
z_1}{4}\right)
to compute\mathrel {\mathop }
\begin{}
\begin{align*}
\in... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005715889856219292,
-0.0007206866866908967,
0.0014432811876758933,
0.03531793877482414,
0.008554755710065365,
0.0049985419027507305,
0.03183804079890251,
0.022527778521180153,
-0.011828609742224216,
0.023825110867619514,
-0.049848053604364395,
-0.01061522401869297,
0.002384800463914871,
... | |
13d697a7cf85c0fce14013371fc4493f46987c56 | subsection | 79 | 108 | Phase Space Transformations | \end{align*}
\end{}
Taking the quotient proves the formula for the covariance matrix.
\end{}
\right.}\end{}
[Proof of Theorem REF ]
Adaptation Axiom REF (A1) follows from
\left(f({\cdot -a)\ast g({\cdot -b)(x)
}}\right.&=
\int f(y-a)\, g(x-y-b)\, \mathrm {d}y
=
\int f(y)\, g(x-(a+b)-y)\, \mathrm {d}y
\\
&=
(f\ast g)(x... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.0013511285651475191,
-0.00341167114675045,
-0.029612313956022263,
-0.011846451088786125,
0.0258440300822258,
0.0018088150536641479,
0.03987974673509598,
0.038262587040662766,
0.020611146464943886,
-0.006117742508649826,
-0.03551647067070007,
0.0018316993955522776,
-0.021862156689167023,
... | |
c41536759f1969d8fd78f43c68a92c67c8d7441c | subsection | 80 | 108 | Phase Space Transformations | The claim follows from the definitions (REF ) of
\mu _f^{(d)} and (REF ) of
\mu _f^{(e)}.
[Proof of Proposition ]
For the h-adaptive convolution of type two the property \left\Vert
G({\cdot ,y)_p = \left\Vert g\right\Vert _p is straightforward for all
y\in \mathbb {R}^d, 1\le p \le \infty and Theorem \ref {theorem\mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03579450771212578,
0.008384092710912228,
-0.010909237898886204,
-0.02456485852599144,
-0.020018070936203003,
-0.027677424252033234,
0.006740078330039978,
0.03478750213980675,
0.03912068158388138,
0.005679669789969921,
-0.04543735831975937,
-0.005008331965655088,
-0.0037705532740801573,
... | |
fcdfe476c2c2cfc218a74f26e5a013c76a8d63ae | subsection | 81 | 108 | Phase Space Transformations | Hölder^{\prime }s inequality yields
\begin{align*}
|\tilde{G}(x,y)|
&\le \left\Vert g_2\right\Vert _p \int |g_1(z-h(y))|^{1/p^{\prime }}\, |g_1(z-h(y))|^{1/p}
\gamma (x,z)\, \mathrm {d}z
\\
&\le \left\Vert g_2\right\Vert _p \left\Vert g_1({\cdot -h(y))^{1/p^{\prime }}_{p^{\prime }}
\left\Vert g_1({\cdot -h(y))^{1/p}\, ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.023305442184209824,
0.013987843878567219,
-0.02129082754254341,
-0.02271021530032158,
-0.027670443058013916,
0.0026613534428179264,
0.020527714863419533,
0.03063131868839264,
0.01092776469886303,
-0.008707108907401562,
-0.049419138580560684,
0.00428105890750885,
-0.03565259650349617,
-0... | |
4f4cfa4d42f18428acf0ee21a6577db6579343f1 | subsection | 82 | 108 | Phase Space Transformations | \end{align*}
\end{}
Therefore \left\Vert \tilde{G}({\cdot ,y)_p \le \left\Vert g_1\right\Vert _1
\left\Vert g_2\right\Vert _p (for all y\in \mathbb {R}^d and 1\le p\le \infty ) also holds for
type three and again Theorem \ref {theorem\mathrel {\mathop }young1} proves the claim.
}
\right.}\right.}\right.\begin{}[Proof o... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
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e238b0249ea07f0cdea0c99077ae111b83e47525 | subsection | 83 | 108 | Phase Space Transformations | Springer, 1997.
R. A. Horn and C. R. Johnson.
Matrix analysis.
Cambridge University Press, 2012.
I. Klebanov.
Axiomatic approach to variable kernel density estimation.
ArXiv e-prints, 2018.
W. Young.
On the multiplication of successions of fourier constants.
Proceedings of the Royal Society of London. Series A, Contai... | {
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