Fast Multipole Methods in Arbitrary Dimensions with Chenhan Yu - - PowerPoint PPT Presentation

▶
fast multipole methods in arbitrary dimensions
SMART_READER_LITE
LIVE PREVIEW

Fast Multipole Methods in Arbitrary Dimensions with Chenhan Yu - - PowerPoint PPT Presentation

Nov 02, 2023 140 likes •510 views

Fast Multipole Methods in Arbitrary Dimensions with Chenhan Yu James Levitt Severin Riez Bill March Bo Xiao GEORGE BIROS padas.ices.utexas.edu Problem statement and contributions FMM for kernel matrices given points in high D

slide-1
SLIDE 1

Fast Multipole Methods in Arbitrary Dimensions

with Chenhan Yu James Levitt Severin Riez Bill March Bo Xiao

GEORGE BIROS

padas.ices.utexas.edu

slide-2
SLIDE 2

Problem statement and contributions

  • FMM for kernel matrices given points in high D
  • FMM for SPD matrices—no points given
  • Four components

  • Matrix permutations to expose low-rank structure — O(N) 

  • Compress blocks — O(N logN)

  • Fast Matvec — O(N)

  • HPC implementation (MPI async + ARM, x86/KNL, GPUs)

2

slide-3
SLIDE 3

Motivation: Kernel classification

3

slide-4
SLIDE 4
slide-5
SLIDE 5

Motivation: arbitrary SPD matrices

  • Hessian matrix for large scale optimization
  • Schur-complement operators for computing

inverse graph Laplacians

  • Factorization of dense covariance matrices

No available geometry information (i.e., points)

5

slide-6
SLIDE 6

Two algorithms

  • ASKIT: Algebraically Skeletonized Kernel

Independent Treecode

  • GOFMM: Geometry Oblivious Fast Multipole

Method

6

slide-7
SLIDE 7

Highlights ASKIT

7

CFD:12B/3D ~ 700 GB Malhotra, Gholami, & B’ SC14

March, Yu, Xiao, B. KDD’15

Kernels: 1B/128D ~ 1TB

slide-8
SLIDE 8

8

Highlights GOFMM

slide-9
SLIDE 9

Achieving O(N logaN) complexity

9

NYSTROM ENSEMBLE NYSTROM HIERARCHICAL MATRICES

slide-10
SLIDE 10

Sparse + low-rank

10

slide-11
SLIDE 11

Hierarchical matrices, basic idea

11

K11 K12 K21 K22

  • <latexit sha1_base64="r/IpZ RUlDGDMOsHvgtPVTYtm+0=">A
CuXicdVFNj9MwEHXD1xK+unDkYlGBEIcqyW6XVuJQAQckhFgkyq60jirHmbRmHSfYDlBZ+X/8Bf4EVzgyTVvBSjCSrac3b8bjN1mtpHVR9L0X Lp85eq1vevhjZu3bt/p79/9YKvGCJiJSlXmNOMWlNQwc9IpOK0N8DJTcJKdv1jnTz6DsbLS792qhrTkCy0LKbhDat7P Oua+Ex cd6G7CyklGWwkNpnJXdGfm2ReT3 cdzSRx1IWsrYhkx2ZLIjGej8TyVLw3beH0TDeHIYTxIaDZPkaDQ+QHA0ikfjQxoPoy4GZBvH8/3elOWVaErQTihu7Vkc1S713DgpFGDXxkKN4/IFnCHUvASb+u4fLX2ITE6LyuDRjnbs3xWel9auygyVO XSXuhWKPik07VknfunpH FOPVS140DLTbvFY2irqJrg2kuDQinVgi4MBJHpmLJDRcO1xAyAxq+iKosuc6fsIKXUq1yKHijnGe2 EL8owVcpl64pWc1N1Ln6Ebr0QjMvQT0xcAbHO5tDY 7XC zcqFb390her4zlv4fzJLhZBi/SwbT51vz98h98oA8JjF5SqbkFTkmMyLIN/KD/CS/gmdBFiyDjxtp0NvW3CMXIrC/Aa7 2Og=</latexit> <latexit sha1_base64="r/IpZ RUlDGDMOsHvgtPVTYtm+0=">A CuXicdVFNj9MwEHXD1xK+unDkYlGBEIcqyW6XVuJQAQckhFgkyq60jirHmbRmHSfYDlBZ+X/8Bf4EVzgyTVvBSjCSrac3b8bjN1mtpHVR9L0X Lp85eq1vevhjZu3bt/p79/9YKvGCJiJSlXmNOMWlNQwc9IpOK0N8DJTcJKdv1jnTz6DsbLS792qhrTkCy0LKbhDat7P Oua+Ex cd6G7CyklGWwkNpnJXdGfm2ReT3 cdzSRx1IWsrYhkx2ZLIjGej8TyVLw3beH0TDeHIYTxIaDZPkaDQ+QHA0ikfjQxoPoy4GZBvH8/3elOWVaErQTihu7Vkc1S713DgpFGDXxkKN4/IFnCHUvASb+u4fLX2ITE6LyuDRjnbs3xWel9auygyVO XSXuhWKPik07VknfunpH FOPVS140DLTbvFY2irqJrg2kuDQinVgi4MBJHpmLJDRcO1xAyAxq+iKosuc6fsIKXUq1yKHijnGe2 EL8owVcpl64pWc1N1Ln6Ebr0QjMvQT0xcAbHO5tDY 7XC zcqFb390her4zlv4fzJLhZBi/SwbT51vz98h98oA8JjF5SqbkFTkmMyLIN/KD/CS/gmdBFiyDjxtp0NvW3CMXIrC/Aa7 2Og=</latexit> <latexit sha1_base64="r/IpZ RUlDGDMOsHvgtPVTYtm+0=">A CuXicdVFNj9MwEHXD1xK+unDkYlGBEIcqyW6XVuJQAQckhFgkyq60jirHmbRmHSfYDlBZ+X/8Bf4EVzgyTVvBSjCSrac3b8bjN1mtpHVR9L0X Lp85eq1vevhjZu3bt/p79/9YKvGCJiJSlXmNOMWlNQwc9IpOK0N8DJTcJKdv1jnTz6DsbLS792qhrTkCy0LKbhDat7P Oua+Ex cd6G7CyklGWwkNpnJXdGfm2ReT3 cdzSRx1IWsrYhkx2ZLIjGej8TyVLw3beH0TDeHIYTxIaDZPkaDQ+QHA0ikfjQxoPoy4GZBvH8/3elOWVaErQTihu7Vkc1S713DgpFGDXxkKN4/IFnCHUvASb+u4fLX2ITE6LyuDRjnbs3xWel9auygyVO XSXuhWKPik07VknfunpH FOPVS140DLTbvFY2irqJrg2kuDQinVgi4MBJHpmLJDRcO1xAyAxq+iKosuc6fsIKXUq1yKHijnGe2 EL8owVcpl64pWc1N1Ln6Ebr0QjMvQT0xcAbHO5tDY 7XC zcqFb390her4zlv4fzJLhZBi/SwbT51vz98h98oA8JjF5SqbkFTkmMyLIN/KD/CS/gmdBFiyDjxtp0NvW3CMXIrC/Aa7 2Og=</latexit> <latexit sha1_base64="r/IpZ RUlDGDMOsHvgtPVTYtm+0=">A CuXicdVFNj9MwEHXD1xK+unDkYlGBEIcqyW6XVuJQAQckhFgkyq60jirHmbRmHSfYDlBZ+X/8Bf4EVzgyTVvBSjCSrac3b8bjN1mtpHVR9L0X Lp85eq1vevhjZu3bt/p79/9YKvGCJiJSlXmNOMWlNQwc9IpOK0N8DJTcJKdv1jnTz6DsbLS792qhrTkCy0LKbhDat7P Oua+Ex cd6G7CyklGWwkNpnJXdGfm2ReT3 cdzSRx1IWsrYhkx2ZLIjGej8TyVLw3beH0TDeHIYTxIaDZPkaDQ+QHA0ikfjQxoPoy4GZBvH8/3elOWVaErQTihu7Vkc1S713DgpFGDXxkKN4/IFnCHUvASb+u4fLX2ITE6LyuDRjnbs3xWel9auygyVO XSXuhWKPik07VknfunpH FOPVS140DLTbvFY2irqJrg2kuDQinVgi4MBJHpmLJDRcO1xAyAxq+iKosuc6fsIKXUq1yKHijnGe2 EL8owVcpl64pWc1N1Ln6Ebr0QjMvQT0xcAbHO5tDY 7XC zcqFb390her4zlv4fzJLhZBi/SwbT51vz98h98oA8JjF5SqbkFTkmMyLIN/KD/CS/gmdBFiyDjxtp0NvW3CMXIrC/Aa7 2Og=</latexit>

= K11 K22

  • +

 0 K12 K21

  • <latexit sha1_base64="MaAHAKwpmXsK4CO80ODxZcw9WHU=">A
C9XicdVJda9RAFJ3EqjV+bfXRl8F kQpLku7W3QdhUR8E Su4t AJy2Ryszt0MokzE+0S8lf6JL76e/w3 v0odqFeSDhz7 knd85NWilpXRj+8fwbOzdv3d69E9y9d/ Bw87eo6+2rI2AiShVaU5SbkFJDRMn YKTygAvUgXH6dnbZf34OxgrS/3FLSpICj7TMpeCO0xNOxcNW4k0qeLirA3YafA6oJSlMJO6SQvujDxvMfNh2kR S5/TkDKG5xAh5uK4XZ8Z6Owf/+W1Gpue6LJn2X9Vc1uDJUE7 XTDXjTqR6OYhr04PhwMDxAcDqLBsE+jXriKLtnE0XTPG7OsFHUB2gnFrT2NwsolDTdOCgWoWluo8Kp8BqcINS/AJs3Kg5Y+w0xG89Lgox1dZa92NLywdlGkyMQp53ZL VfwTSdLyrJ2LaV2+TBp K5qB1qsv5fXirqSLpdDM2lAOLVAwIWRODIVc264cLjCgBnQ8EOURcF1ts9yXki1yCDntXINs/kG4h0t4I+gZ27esIobqTN0o23QCKy9A/TFwEc 7lMFhjtcPrNypt m9Q7Q80tj6f/BJO6NetHnuDt+szF/lzwhT8kLEpFXZEzekyMyIcLb8fa9A6/vn/sX/k/ 15rqe5uex2Qr/N9/AWX95/Y=</latexit> <latexit sha1_base64="MaAHAKwpmXsK4CO80ODxZcw9WHU=">A C9XicdVJda9RAFJ3EqjV+bfXRl8F kQpLku7W3QdhUR8E Su4t AJy2Ryszt0MokzE+0S8lf6JL76e/w3 v0odqFeSDhz7 knd85NWilpXRj+8fwbOzdv3d69E9y9d/ Bw87eo6+2rI2AiShVaU5SbkFJDRMn YKTygAvUgXH6dnbZf34OxgrS/3FLSpICj7TMpeCO0xNOxcNW4k0qeLirA3YafA6oJSlMJO6SQvujDxvMfNh2kR S5/TkDKG5xAh5uK4XZ8Z6Owf/+W1Gpue6LJn2X9Vc1uDJUE7 XTDXjTqR6OYhr04PhwMDxAcDqLBsE+jXriKLtnE0XTPG7OsFHUB2gnFrT2NwsolDTdOCgWoWluo8Kp8BqcINS/AJs3Kg5Y+w0xG89Lgox1dZa92NLywdlGkyMQp53ZL VfwTSdLyrJ2LaV2+TBp K5qB1qsv5fXirqSLpdDM2lAOLVAwIWRODIVc264cLjCgBnQ8EOURcF1ts9yXki1yCDntXINs/kG4h0t4I+gZ27esIobqTN0o23QCKy9A/TFwEc 7lMFhjtcPrNypt m9Q7Q80tj6f/BJO6NetHnuDt+szF/lzwhT8kLEpFXZEzekyMyIcLb8fa9A6/vn/sX/k/ 15rqe5uex2Qr/N9/AWX95/Y=</latexit> <latexit sha1_base64="MaAHAKwpmXsK4CO80ODxZcw9WHU=">A C9XicdVJda9RAFJ3EqjV+bfXRl8F kQpLku7W3QdhUR8E Su4t AJy2Ryszt0MokzE+0S8lf6JL76e/w3 v0odqFeSDhz7 knd85NWilpXRj+8fwbOzdv3d69E9y9d/ Bw87eo6+2rI2AiShVaU5SbkFJDRMn YKTygAvUgXH6dnbZf34OxgrS/3FLSpICj7TMpeCO0xNOxcNW4k0qeLirA3YafA6oJSlMJO6SQvujDxvMfNh2kR S5/TkDKG5xAh5uK4XZ8Z6Owf/+W1Gpue6LJn2X9Vc1uDJUE7 XTDXjTqR6OYhr04PhwMDxAcDqLBsE+jXriKLtnE0XTPG7OsFHUB2gnFrT2NwsolDTdOCgWoWluo8Kp8BqcINS/AJs3Kg5Y+w0xG89Lgox1dZa92NLywdlGkyMQp53ZL VfwTSdLyrJ2LaV2+TBp K5qB1qsv5fXirqSLpdDM2lAOLVAwIWRODIVc264cLjCgBnQ8EOURcF1ts9yXki1yCDntXINs/kG4h0t4I+gZ27esIobqTN0o23QCKy9A/TFwEc 7lMFhjtcPrNypt m9Q7Q80tj6f/BJO6NetHnuDt+szF/lzwhT8kLEpFXZEzekyMyIcLb8fa9A6/vn/sX/k/ 15rqe5uex2Qr/N9/AWX95/Y=</latexit> <latexit sha1_base64="MaAHAKwpmXsK4CO80ODxZcw9WHU=">A C9XicdVJda9RAFJ3EqjV+bfXRl8F kQpLku7W3QdhUR8E Su4t AJy2Ryszt0MokzE+0S8lf6JL76e/w3 v0odqFeSDhz7 knd85NWilpXRj+8fwbOzdv3d69E9y9d/ Bw87eo6+2rI2AiShVaU5SbkFJDRMn YKTygAvUgXH6dnbZf34OxgrS/3FLSpICj7TMpeCO0xNOxcNW4k0qeLirA3YafA6oJSlMJO6SQvujDxvMfNh2kR S5/TkDKG5xAh5uK4XZ8Z6Owf/+W1Gpue6LJn2X9Vc1uDJUE7 XTDXjTqR6OYhr04PhwMDxAcDqLBsE+jXriKLtnE0XTPG7OsFHUB2gnFrT2NwsolDTdOCgWoWluo8Kp8BqcINS/AJs3Kg5Y+w0xG89Lgox1dZa92NLywdlGkyMQp53ZL VfwTSdLyrJ2LaV2+TBp K5qB1qsv5fXirqSLpdDM2lAOLVAwIWRODIVc264cLjCgBnQ8EOURcF1ts9yXki1yCDntXINs/kG4h0t4I+gZ27esIobqTN0o23QCKy9A/TFwEc 7lMFhjtcPrNypt m9Q7Q80tj6f/BJO6NetHnuDt+szF/lzwhT8kLEpFXZEzekyMyIcLb8fa9A6/vn/sX/k/ 15rqe5uex2Qr/N9/AWX95/Y=</latexit>
slide-12
SLIDE 12

Constructing the approximation

12

slide-13
SLIDE 13

Idea I: far-field —> low rank

13

xw xj xi

slide-14
SLIDE 14

Idea II: Near/Far field split

14

slide-15
SLIDE 15

Idea III: recursion

15

1 2 3 4

slide-16
SLIDE 16

Challenges in high-dimensions

  • Constructing the far-field approximations 


polynomial in ambient-D

  • Near-far field decomposition


polynomial in ambient-D

  • No scalable algorithms (other than Nystrom)
  • Nystrom method assumes low rank


provably not the case with increasing N

16

slide-17
SLIDE 17

Randomized linear algebra — Nystrom method

  • Low-rank decomposition of G
  • Random sampling of O(s) points, s: target rank

  • Work
  • Error

17

slide-18
SLIDE 18

ASKIT

  • Randomized Linear Algebra — far field approximation
  • Parallel binary trees — permutation, partitioning
  • Nearest neighbors — pruning and sampling
  • Treecode / FMM
  • MPI / OpenMP / SIMD / GPU acceleration
  • Inspired by


Ying & B. & Zorin’03
 Haiko & Martinsson & Tropp’11
 Drineas & Kahan & Mahoney'06

18

SISC’15,16 ACHA’15 KDD’15 SC’15 IPDPS’15,16,17

slide-19
SLIDE 19

Far-field s-rank approximation

  • SVD is too expensive — use sampling
  • Sample rows 


leverage, norm, range-space

  • Interpolative decomposition
  • ASKIT: approximate norm adaptive sampling

using nearest-neighbors + adaptive rank selection

19

slide-20
SLIDE 20

Skeletonization (far field approximation)

20

slide-21
SLIDE 21

Evaluation

21

slide-22
SLIDE 22

Evaluation

22

slide-23
SLIDE 23

Complexity and error

  • Work
  • Error
  • Nystrom

23

  • ff-diagonal

diagonal

slide-24
SLIDE 24

Parallel complexity

24

Points per MPI task n = N

p

Tree depth D = log N

s

Tree construction ≤ (ts + tw) log2p log N + (tw log p) (d + k) n Skeletonization ≤ tf ⇣n s + log p ⌘ s3 Evaluation ≤ tsp + (tw + tf)d k s D n

slide-25
SLIDE 25

Summary of ASKIT features

  • Binary tree for matrix permutation
  • Approximate randomized nearest neighbors
  • Nearest neighbors for skeletonization
  • Bottom-up recursive low-rank approximation
  • Top-down pass for fast evaluation
  • Adaptive sampling and rank selection

25

slide-26
SLIDE 26

Gaussian

26

3D, 1M points 64D/20D intr, 1M points

slide-27
SLIDE 27

Kernel acceleration

27

slide-28
SLIDE 28

Nystrom vs ASKIT (8M/784D)

28

NYSTROM ASKIT

slide-29
SLIDE 29

Kernel regression scaling

29

MNIST dataset for OCR strong scaling, 8M points d=784

slide-30
SLIDE 30

Related work

  • 2D & 3D N-body 


Barnes & Hut, Greengard & Rokhlin, Darve, Hackbush / Novak / Bebendorf

  • High-D 


Griebel, Duraiswami, Vuduc / Gray, Kondor, Mahoney & Darve
 March / Bo / Biros [ASKIT]

  • Purely algebraic


Li et al / STRUMPACK
 Darve et al / HODLR

Kij = K(xi, xj) = K(xi − xj)

<latexit sha1_base64="+hDiQfJFUKXT5dN9+sGRM4wecOY=">A CUXicdVBNb9QwEHXCV0mBbuHIxWKFVB ESeiW3QNSBRekXorE0kqbKJo4k9at7QTbga6i/S38Gq5w4cJf4YSzDRKVYCRLb968+fArGsGNjaKfn /t+o2btzZuB5t37t7bGm3f/2DqVjOcs1rU+rgAg4IrnFtuBR43GkEWAo+K8zd9/egTasNr9d4uG8wknChecQbWUfloljaguSpRWRo1NgiCdHGQd/xs9Yoe7Fzk/NlFfvaEDgl9Tvs0zZwuH42jMJ7txrOERmGS7E2mLxzYm8ST6S6Nw2gdYzLEYb7teWlZs1a6XUyAMYvYbcw60JYzgasgbQ02wM7hB cOKpBosm79x V97JiSVrV2z926Zv/u6EAas5SFU0qwp+bKtErgR5X1kr72T0lrq2nWcdW0FhW73Fe1gtqa9r7RkmtkViwdAKa5O5myU9DArHM3SDUq/MxqKUGVT9MKJBfLEitohe1SUw1w5Uz74wz9P5gn4SyM3yXj/deDexvkIXlEdkhMXpJ98pYckjlh5Av5Sr6R794P75dPfP9S6ntDzwNyJfzN36FfsHg=</latexit> <latexit sha1_base64="+hDiQfJFUKXT5dN9+sGRM4wecOY=">A CUXicdVBNb9QwEHXCV0mBbuHIxWKFVB ESeiW3QNSBRekXorE0kqbKJo4k9at7QTbga6i/S38Gq5w4cJf4YSzDRKVYCRLb968+fArGsGNjaKfn /t+o2btzZuB5t37t7bGm3f/2DqVjOcs1rU+rgAg4IrnFtuBR43GkEWAo+K8zd9/egTasNr9d4uG8wknChecQbWUfloljaguSpRWRo1NgiCdHGQd/xs9Yoe7Fzk/NlFfvaEDgl9Tvs0zZwuH42jMJ7txrOERmGS7E2mLxzYm8ST6S6Nw2gdYzLEYb7teWlZs1a6XUyAMYvYbcw60JYzgasgbQ02wM7hB cOKpBosm79x V97JiSVrV2z926Zv/u6EAas5SFU0qwp+bKtErgR5X1kr72T0lrq2nWcdW0FhW73Fe1gtqa9r7RkmtkViwdAKa5O5myU9DArHM3SDUq/MxqKUGVT9MKJBfLEitohe1SUw1w5Uz74wz9P5gn4SyM3yXj/deDexvkIXlEdkhMXpJ98pYckjlh5Av5Sr6R794P75dPfP9S6ntDzwNyJfzN36FfsHg=</latexit> <latexit sha1_base64="+hDiQfJFUKXT5dN9+sGRM4wecOY=">A CUXicdVBNb9QwEHXCV0mBbuHIxWKFVB ESeiW3QNSBRekXorE0kqbKJo4k9at7QTbga6i/S38Gq5w4cJf4YSzDRKVYCRLb968+fArGsGNjaKfn /t+o2btzZuB5t37t7bGm3f/2DqVjOcs1rU+rgAg4IrnFtuBR43GkEWAo+K8zd9/egTasNr9d4uG8wknChecQbWUfloljaguSpRWRo1NgiCdHGQd/xs9Yoe7Fzk/NlFfvaEDgl9Tvs0zZwuH42jMJ7txrOERmGS7E2mLxzYm8ST6S6Nw2gdYzLEYb7teWlZs1a6XUyAMYvYbcw60JYzgasgbQ02wM7hB cOKpBosm79x V97JiSVrV2z926Zv/u6EAas5SFU0qwp+bKtErgR5X1kr72T0lrq2nWcdW0FhW73Fe1gtqa9r7RkmtkViwdAKa5O5myU9DArHM3SDUq/MxqKUGVT9MKJBfLEitohe1SUw1w5Uz74wz9P5gn4SyM3yXj/deDexvkIXlEdkhMXpJ98pYckjlh5Av5Sr6R794P75dPfP9S6ntDzwNyJfzN36FfsHg=</latexit> <latexit sha1_base64="+hDiQfJFUKXT5dN9+sGRM4wecOY=">A CUXicdVBNb9QwEHXCV0mBbuHIxWKFVB ESeiW3QNSBRekXorE0kqbKJo4k9at7QTbga6i/S38Gq5w4cJf4YSzDRKVYCRLb968+fArGsGNjaKfn /t+o2btzZuB5t37t7bGm3f/2DqVjOcs1rU+rgAg4IrnFtuBR43GkEWAo+K8zd9/egTasNr9d4uG8wknChecQbWUfloljaguSpRWRo1NgiCdHGQd/xs9Yoe7Fzk/NlFfvaEDgl9Tvs0zZwuH42jMJ7txrOERmGS7E2mLxzYm8ST6S6Nw2gdYzLEYb7teWlZs1a6XUyAMYvYbcw60JYzgasgbQ02wM7hB cOKpBosm79x V97JiSVrV2z926Zv/u6EAas5SFU0qwp+bKtErgR5X1kr72T0lrq2nWcdW0FhW73Fe1gtqa9r7RkmtkViwdAKa5O5myU9DArHM3SDUq/MxqKUGVT9MKJBfLEitohe1SUw1w5Uz74wz9P5gn4SyM3yXj/deDexvkIXlEdkhMXpJ98pYckjlh5Av5Sr6R794P75dPfP9S6ntDzwNyJfzN36FfsHg=</latexit>
slide-31
SLIDE 31

Geometry oblivious FMM

  • Permute matrix to expose low-rank structure
  • Geometry-based algorithms: need distance

between indices i,j

  • Gram vectors


distance(i,j) = function(Kii, Kij, Kjj)


slide-32
SLIDE 32

Geometry oblivious FMM

  • Gram vectors (K is SPD):
  • Distances

32

Kij = φi · φj

<latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="XJg4 bHi0XuIhMH7Qi5i86MvklE=">A C HicbZC/TsMwEMad8q+UAu3MElEhIY qYQE2JBbGIhFaiUTVxbm0Vm0n2A6oivoCrCw8DRPiHXgb3DYDpZxk6afvPp/PX5xzpo3nfTu1jc2t7Z36bmOv2dg/OGw1H3RWKIoBzXimBjFo5ExiYJjhOMgVgog59uPJzbzf 0alWSbvzT HSMBIspR MFbqDVsdr+styl0Hv4IOqWrYdpw yWghUBrKQetH38tNVI yjHKcNcJCYw50AiN8tChBoI7KxZ4z98QqiZtmyh5p3IX6+0YJQu piK1TgBnrlWkpxycZzS3z3r+WwqSXUclkXhiUdPleWnDXZO78727CF LDpxaAKmZXdukYF BjE2qECiW+0EwIkMlZmIJgfJpgCgU3ZajTCmc2M/9vQusQnHevuv6dR+rkiByTU+KTC3JNbkmPBIS hLySN+fd+XA+l9HWnCrjNlkp5+sHkL+dJw= </latexit> <latexit sha1_base64="UKAtfIfQzOqzGjwbmOBHZs3EGX0=">A CQ3icbVA9bxQxEJ0NX8E cqGlsYiQEMVpl4KQAgmJBokmSByJdF6tZr2zOSf+WGwv6LS6f8KvSRsaSn4JFQLfR0EIT7L0P Gb/zqTqsQ8/xHtnXj5q3bd7bvsns79x/sjvZ2PgbXe0kT6bTzJzUG0srSJKqo6aTzhKbWdFyfv1n2jz+TD8rZD3HeUWnw1KpWSYypVI1eiA69sg3ZyPMuMiamnL2rBnW24K+46GaqUlzIxsX15YyJkjFWjfbzcb4Cv06KDdmHDY6qvSwTjZO9SUZSYwjTItmVA/qopKYFE32gDuU5ntI0UYuGQjmsPrjgT1Kl4a3z6aRFV9W/JwY0IcxNnZQG4yxcea3V9MmWS8my919JH9uX5aBs10eycu3X9p Hx5eh8UZ5klHPE0HpV qZyxl6lDF y4QnS1+kMwZt80y0aJSeN9Rir+MgQruhixRa8W9E18nk+fhwXLzPYRsewWN4CgUcwGt4C0cwAQlf4QIu4Vv2PfuZ/Vqnu5VtYn4IV5D9/gMuTrFa</latexit> <latexit sha1_base64="9auKeIDfznjBaWfs AsxmAS3zac=">A CQ3icdVDPaxQxFM60amusduvVS7AI4mGYWbvr7kEQehG8VHBtYTM bzJvum TzJhkKsuw/0n/ml714tG/xJNo9odgQT8IfHnve/levqJR0vk +R5tbd+5e29n9z59sPfw0X7vYO+jq1srcCJqVduzAhwqaXDipVd41lgEXSg8LS6Pl/3TK7RO1uaDnzeYaTg3spICfCjlvSFvwEpTovEsaTylfMrou7yTFwv2mvFmJnPJuChrv75cUJ5RSvPeYRKn46N03GdJ3O8PB6OXgQwH6WB0xNI4WeGQbHCSH0QRL2vR6mAkFDg3TYNd1oH1UihcUN46bEBcwjlOAzWg0WXd6oML9ixUSlbVNpyw6Kr690QH2rm5LoJSg5+5W69VCj+ZbClZ9v4paX01yjp mtajEWu/qlXM12wZGiulReHVPBAQVoaVmZiB eFDtJRbNPhZ1FqDKV/wCrRU8xIraJXvuKs2dBFC+5M +z+Z9ONxnL5PyC5 Qp6S5yQlr8gb8packAkR5JrckC/ka/Qt+hH9XKe7FW1ifkxuIfr1G67Esas=</latexit> <latexit sha1_base64="NBzBMyQ7Kw5XtK P17l6zP73qHw=">A CTnicdVDLbhMxFPWEAsG80rJkYzVCQixGM2kTk WlCjZIbIpEaKV4NPJ47jRu/RhsT6tolD/ha9jChiVfwgqB80CiEhzJ0vG5 /pen6KWwvk +R51bu3cvnO3ew/f /Dw0ePe7t4HZxrLYcqN PasYA6k0D 1wks4qy0wVUg4LS5fr+qnV2CdMPq9X9SQKXauRSU480HKeyNaMyt0CdqTpPY 0xnBb/NWXCzJEaH1XOSCUF4av7lcYJphjPNeP4nTyWE6GZAkHgxGw/FBIKNhOhwfkjRO1uijLU7y3Si peGNCoO4ZM7N0jAua5n1gktY to4qBm/ZOcwC1QzBS5r1x9ckmdBKUl bDh 0bX6d0fLlHMLVQSnYn7ubrxWSfios5VlVfunpfHVOGuFrhsPm /mVY0k3pBVaKQUFriXi0AYtyKsTPicWcZ9iBZTCxqu VGK6fIFrZgSclFCxRrpW+q LV2G0P4kQ/5PpoN4Eqfvkv7xq216XfQU7aPnKEUv0TF6g07QFH 0CX1GX9DX6Fv0I/oZ/dpYO9G25wm6gU73N1v4smE=</latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit> <latexit sha1_base64="6RbfIkZJiEHCaoq423T/gMl8ovE=">A CTnicdVDLihQxFE21rza+enTpJtgI4qKoqplu xfCoBvBzQi2M9ApilTq1nRm8i T1AxN0X/i17jVjUu/xJVo+iE4oAcCJ+e m3tzykYK5 Pke9S7dv3GzVv92/jO3Xv3Hwz2Hn5wprUcZtxIY09K5kAKDTMv ISTxgJTpYTj8vz1un58AdYJo9/7ZQO5Yqda1I zH6RiMKYNs0JXoD1JGo8xnRP8tujE2Yq8JLRZiEIQyivjt5czTHOMcTEYJnE6PUinGUniLBuPJvuBjEfpaHJA0j ZYIh2OCr2o hWhrcqDOKSOTdPw7i8Y9YL mGFaeugYfycncI8UM0UuLzbfHBFngalIrWx4YRFN+rfHR1Tzi1VGZyK+YW78lot4aPO15Z17Z+W1teTvBO6aT1ovp1Xt5J4Q9ahkUpY4F4uA2HcirAy4QtmGfchWkwtaLjkRim q+e0ZkrIZQU1a6XvqKt3dBVC+5M +T+Z fE0Tt9lw8NXu/T6 DF6gp6hFL1Ah+gNOkIzxNEn9Bl9QV+jb9GP6Gf0a2vtRbueR+gKev3fXJiyYw= </latexit>

Euclidean kφi φjk2

2

Kii + Kjj 2Kij Angle sin ⇣

φi·φj kφik2kφjk2

⌘ 1

Kij2 KiiKjj

<latexit sha1_base64="S5Ot70zc8rVC6Uh0cpNGVIH/i1w=">A DFnicdVJNb9QwE 3CVwlfWzhysdhSFRCrJHTL7q2AkJB6KRJLK9XbleNMdr1 nGA7oFWaP8AVLvwaTogrV/4Mwk5SRCUYKfLzm+eZ8XPigjOlg+Cn6124eOnylbWr/rXrN27e6q3f qvyUlKY0Jzn8jAmCjgTMNFMczgsJ As5nAQn7yw+YP3IBXLxRu9KmCakblgKaNEG2rW+4ULIplIQGgUFNr3cQxzJipN4pITWVec0tp/WVLOEiACbaINfIpwsWAzh 63YInw6Sw6j Zsdm9WMVY/MstyWRtFZAmDNhDG/jMx54BsbFpCMYE5pHoL4VQSWnVlMU1y3ZWuK1O85W2Ts83SbmqEJZsv9IOmcWjHacrs2YbHUV21s7Sj1M0AGETy526+7896/WAQjrfDcYSCQRTtDEdPDNgZhsPRNgoHQRN9p4v92br 4iSnZWYMo5wodRQa26YVkZpRDrWPSwUFoSdkDkcGCpKBmlbNQ9XovmESlObSfMbwhv37REUypVZ bJQZ0Qt1rlrK4Z2YWonN/VNS6nQ0rZgoSg2Ctv3SkiOdI/v4KGESqOYrAwiVzIyM6I Yz7T5RXwsQcAHm cZEclDnJKM8VUCKSm5r BKO1gb086cQf8Hk2gwHoSvo/7u8 69Ne uc8/ZckLnqbPrvHL2nYlD3cT96H5yP3tfvK/eN+97K/Xc7swd51x4P34DZK729Q= </latexit> <latexit sha1_base64="S5Ot70zc8rVC6Uh0cpNGVIH/i1w=">A DFnicdVJNb9QwE 3CVwlfWzhysdhSFRCrJHTL7q2AkJB6KRJLK9XbleNMdr1 nGA7oFWaP8AVLvwaTogrV/4Mwk5SRCUYKfLzm+eZ8XPigjOlg+Cn6124eOnylbWr/rXrN27e6q3f qvyUlKY0Jzn8jAmCjgTMNFMczgsJ As5nAQn7yw+YP3IBXLxRu9KmCakblgKaNEG2rW+4ULIplIQGgUFNr3cQxzJipN4pITWVec0tp/WVLOEiACbaINfIpwsWAzh 63YInw6Sw6j Zsdm9WMVY/MstyWRtFZAmDNhDG/jMx54BsbFpCMYE5pHoL4VQSWnVlMU1y3ZWuK1O85W2Ts83SbmqEJZsv9IOmcWjHacrs2YbHUV21s7Sj1M0AGETy526+7896/WAQjrfDcYSCQRTtDEdPDNgZhsPRNgoHQRN9p4v92br 4iSnZWYMo5wodRQa26YVkZpRDrWPSwUFoSdkDkcGCpKBmlbNQ9XovmESlObSfMbwhv37REUypVZ bJQZ0Qt1rlrK4Z2YWonN/VNS6nQ0rZgoSg2Ctv3SkiOdI/v4KGESqOYrAwiVzIyM6I Yz7T5RXwsQcAHm cZEclDnJKM8VUCKSm5r BKO1gb086cQf8Hk2gwHoSvo/7u8 69Ne uc8/ZckLnqbPrvHL2nYlD3cT96H5yP3tfvK/eN+97K/Xc7swd51x4P34DZK729Q= </latexit> <latexit sha1_base64="S5Ot70zc8rVC6Uh0cpNGVIH/i1w=">A DFnicdVJNb9QwE 3CVwlfWzhysdhSFRCrJHTL7q2AkJB6KRJLK9XbleNMdr1 nGA7oFWaP8AVLvwaTogrV/4Mwk5SRCUYKfLzm+eZ8XPigjOlg+Cn6124eOnylbWr/rXrN27e6q3f qvyUlKY0Jzn8jAmCjgTMNFMczgsJ As5nAQn7yw+YP3IBXLxRu9KmCakblgKaNEG2rW+4ULIplIQGgUFNr3cQxzJipN4pITWVec0tp/WVLOEiACbaINfIpwsWAzh 63YInw6Sw6j Zsdm9WMVY/MstyWRtFZAmDNhDG/jMx54BsbFpCMYE5pHoL4VQSWnVlMU1y3ZWuK1O85W2Ts83SbmqEJZsv9IOmcWjHacrs2YbHUV21s7Sj1M0AGETy526+7896/WAQjrfDcYSCQRTtDEdPDNgZhsPRNgoHQRN9p4v92br 4iSnZWYMo5wodRQa26YVkZpRDrWPSwUFoSdkDkcGCpKBmlbNQ9XovmESlObSfMbwhv37REUypVZ bJQZ0Qt1rlrK4Z2YWonN/VNS6nQ0rZgoSg2Ctv3SkiOdI/v4KGESqOYrAwiVzIyM6I Yz7T5RXwsQcAHm cZEclDnJKM8VUCKSm5r BKO1gb086cQf8Hk2gwHoSvo/7u8 69Ne uc8/ZckLnqbPrvHL2nYlD3cT96H5yP3tfvK/eN+97K/Xc7swd51x4P34DZK729Q= </latexit> <latexit sha1_base64="S5Ot70zc8rVC6Uh0cpNGVIH/i1w=">A DFnicdVJNb9QwE 3CVwlfWzhysdhSFRCrJHTL7q2AkJB6KRJLK9XbleNMdr1 nGA7oFWaP8AVLvwaTogrV/4Mwk5SRCUYKfLzm+eZ8XPigjOlg+Cn6124eOnylbWr/rXrN27e6q3f qvyUlKY0Jzn8jAmCjgTMNFMczgsJ As5nAQn7yw+YP3IBXLxRu9KmCakblgKaNEG2rW+4ULIplIQGgUFNr3cQxzJipN4pITWVec0tp/WVLOEiACbaINfIpwsWAzh 63YInw6Sw6j Zsdm9WMVY/MstyWRtFZAmDNhDG/jMx54BsbFpCMYE5pHoL4VQSWnVlMU1y3ZWuK1O85W2Ts83SbmqEJZsv9IOmcWjHacrs2YbHUV21s7Sj1M0AGETy526+7896/WAQjrfDcYSCQRTtDEdPDNgZhsPRNgoHQRN9p4v92br 4iSnZWYMo5wodRQa26YVkZpRDrWPSwUFoSdkDkcGCpKBmlbNQ9XovmESlObSfMbwhv37REUypVZ bJQZ0Qt1rlrK4Z2YWonN/VNS6nQ0rZgoSg2Ctv3SkiOdI/v4KGESqOYrAwiVzIyM6I Yz7T5RXwsQcAHm cZEclDnJKM8VUCKSm5r BKO1gb086cQf8Hk2gwHoSvo/7u8 69Ne uc8/ZckLnqbPrvHL2nYlD3cT96H5yP3tfvK/eN+97K/Xc7swd51x4P34DZK729Q= </latexit>
slide-33
SLIDE 33

33

slide-34
SLIDE 34

GOFMM vs Others

slide-35
SLIDE 35

GOFMM vs Others

35

slide-36
SLIDE 36

Summary

  • ASKIT for arbitrary dimension FMM using

geometry

  • Geometry Oblivious Fast Multipole Method for

sketching dense SPD matrices; Gram distances

36