SOLVING SPARSE REPRESENTATIONS FOR OBJECT CLASSIFICATION USING QUANTUM D-WAVE 2X MACHINE
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1 SOLVING SPARSE REPRESENTATIONS FOR OBJECT CLASSIFICATION USING QUANTUM D-WAVE 2X MACHINE! Nga Nguyen, Amy Larson, Carleton Coffrin, John Perry, Gary Salazar, and Garrett Kenyon Los Alamos National Laboratory & New Mexico Consortium 5th Neuro Inspired Computational Elements 2017, San Jose, Mar. 6-8, 2017
2 OUTLINE A. SPARSE CODING ON A QUANTUM D-WAVE B. CHOOSING DATASET C. IMPLEMENTATION ON D-WAVE MACHINE D. COMPARISON WITH CLASSICAL SOLVER E. COMPRESSIVE SENSING F. SUMMARY AND FUTURE WORK!
3 Quantum D-Wave machine 2X: a quantum annealer A. METHODOLOGY Solving a sparse-coding (SC) problem Objective function is of the form: reconstruction error Lp-sparseness penalty Olshausen and Field, Nature 381, 607 (1996) Rozell, Johnson, Baraniuk, and Olshausen, Neur. Comp. 20, 2526 (2008)
4 Quantum D-Wave machine 2X: a quantum annealer A. METHODOLOGY Solving a sparse-coding (SC) problem Objective function is of the form: reconstruction error Lp-sparseness penalty Olshausen and Field, Nature 381, 607 (1996) Rozell, Johnson, Baraniuk, and Olshausen, Neur. Comp. 20, 2526 (2008) non-convex problem NP-hard class
5 Quantum D-Wave machine 2X: a quantum annealer A. METHODOLOGY Solving a sparse-coding (SC) problem Objective function is of the form: reconstruction error Lp-sparseness penalty an example of SC reconstruction courtesy of Xinhua Zhang
6 Quantum D-Wave machine 2X: a quantum annealer A. METHODOLOGY mapping the sparse-coding problem onto a quantum unconstrained binary optimization (QUBO): D-Wave Hamiltonian: 8 Nb qubits built from 16 couplers where D-Wave
7 Quantum D-Wave machine 2X: a quantum annealer A. METHODOLOGY mapping the sparse-coding problem onto a quantum unconstrained binary optimization (QUBO): D-Wave Hamiltonian: 8 Nb qubits built from 16 couplers where This mapping is achieved by the relations: D-Wave analogous to L0-sparseness penalty [Nguyen and Kenyon, PMES-16 (2016)]
8 A. METHODOLOGY 4 row qubits
9 A. METHODOLOGY 4 column qubits
10 A. METHODOLOGY unit cell
11 A. METHODOLOGY intra-cell couplings
12 A. METHODOLOGY intra-cell couplings
13 A. METHODOLOGY neighboring couplings
14 A. METHODOLOGY neighboring couplings
15 full Chimera: 12 x 12 x 8
16 B. DATASET CIFAR-10 airplane!! automobile!! ship!! truck 32x32 24x24 edge detection
17 C. IMPLEMENTATION ON D-WAVE MACHINE 8 hand-designed features row orthogonality!! number of features column
18 C. IMPLEMENTATION ON D-WAVE MACHINE 8 hand-designed features row orthogonality!! number of features complete set: column
19 C. IMPLEMENTATION ON D-WAVE MACHINE 8 hand-designed features row column complete set: recon orig orthogonality!! number of features
20 C. IMPLEMENTATION ON D-WAVE MACHINE 8 hand-designed features row column orthogonality!! number of features complete set: Desire: Randomly generated 8 < Nf < 1152 :
21 Building features
22 8 and 32 features 24x24 patch images airplane automobile ship truck 1100 active qubits! 3068 coupling strengths overcomplete order: original recon! 8x12x12 stride: 2, 4 recon! 32x6x6
23 32 and 1152 features 24x24 patch images airplane automobile ship truck 1100 active qubits! 3068 coupling strengths overcomplete order: stride: 4, 24 original recon! non-conv recon! 32x6x6
24 RESULTS 12x12 patch images ship truck original airplane automobile overcomplete order : 8 recon stride: 4 Classification task: SVM (liblinear) 1042 training/208 test images classes air auto bird cat deer dog frog horse ship accur % 93.38% 90.87% 89.42% 94.71% 88.94% 87.98% 89.9% (binary) truck 89.9% 85.58% Nguyen and Kenyon, PMES-16 (2016)
25 D. COMPARISON WITH A CLASSICAL SOLVER!! So far, quantum computation (D-Wave 2X) has NOT outperformed its classical counterpart (GUROBI). Both are comparable. We already made the problem hard. We need to make it harder. How can we make the SC problem harder for both?
26 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity
27 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity EMBEDDING technique (c.f. D-Wave documents)
28 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity EMBEDDING technique (c.f. D-Wave documents)
29 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity EMBEDDING technique (c.f. D-Wave documents) mapping
30 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity EMBEDDING technique Employ all bipartite couplings! Small number of nodes (qubits) 5x5
31 D. COMPARISON WITH A CLASSICAL SOLVER From SC perspective: more overcomplete, harder to solve Meanwhile: The full Chimera in D-Wave offers a certain set of (nearest-neighbor) connectivity EMBEDDING technique 5x5 Employ all bipartite couplings! Small number of nodes (qubits) In practice (D-Wave 2X): Fully connected: 49 nodes Partially orthogonal: 74 nodes Feature optimization!
32 D. COMPARISON WITH A CLASSICAL SOLVER Feature Learning (in progress) before after 5x5 5x5
33 D. COMPARISON WITH A CLASSICAL SOLVER STARTING TO SEE SOMETHING GOOD No. of (random) Hamiltonians: 1 problem solver 49 nodes:! fully connected GUROBI D-Wave 2X (best classical solver) (ISING) Energy Time Energy Time ~ 480 seconds few seconds 72 nodes:! partially! Chimera-orthogonal Energy Time min Energy Time few seconds
34 D. COMPARISON WITH A CLASSICAL SOLVER STARTING TO SEE SOMETHING GOOD No. of (random) Hamiltonians: 1 problem solver 49 nodes:! fully connected GUROBI (best classical solver) Energy Time ~ 480 seconds D-Wave 2X Energy Time (ISING) few seconds 72 nodes:! partially! Chimera-orthogonal Energy Time Energy ~1800 seconds Time few seconds
35 COMPRESSIVE SENSING ON A QUANTUM MACHINE WHY NOT?
36 COMPRESSIVE SENSING ON A QUANTUM MACHINE WHY NOT? COMPRESSIVE SENSING in a nutshell y = x y: measurement, sparse! x: compressible signal! : sensing matrix
37 COMPRESSIVE SENSING ON A QUANTUM MACHINE WHY NOT? COMPRESSIVE SENSING in a nutshell y = x L0: NP-hard L1: mostlyused y: measurement, sparse! x: compressible signal! : sensing matrix wide applications particularly in image processing (X-ray, CT, ), sampling, etc (c.f. Candes, Baraniuk, Compressive Sensing)
38 COMPRESSIVE SENSING ON A QUANTUM MACHINE Radiograph Experiment Abel transform algorithm Obtaining radial density profile f(r) of a cylindrically symmetric object Charge coupled device camera object scintillator X-rays cross-section
39 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm projected radial density of a cylindrically symmetric object: h(y) projected, measurable
40 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm projected radial density of a cylindrically symmetric object: h(y) projected, measurable Task: Recovering the density profile of the cylindrical object f(r):
41 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm projected radial density of a cylindrically symmetric object: h(y) projected, measurable Task: Recovering the density profile of the cylindrical object f(r): y How?: one method:! to obtain a slice of the R r x original object Abel inversion h(y) = A * f(r)
42 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm + D-Wave QUBO Our method is applicable to e.g. X-ray images as sparse inputs h = A f We obtain:
43 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm + D-Wave QUBO Our method is applicable to e.g. X-ray images as sparse inputs h = A f Next: to choose (learn) domain basis: We obtain: are sparse Fourier coefficients.
44 COMPRESSIVE SENSING ON A QUANTUM MACHINE Abel transform algorithm + D-Wave QUBO Our method is applicable to e.g. X-ray images as sparse inputs h = A f Next: to choose (learn) domain basis: We obtain: are sparse Fourier coefficients. then: h = A F s Ising! input
45 COMPRESSIVE SENSING ON A QUANTUM MACHINE Results D-Wave Ising pulls out ALL non-zero Fourier coefficients (frequencies)
46 COMPRESSIVE SENSING ON A QUANTUM MACHINE Results solver problem Compressive! sensing on! radiograph! images GUROBI energy time ~ cutoff 9 hrs D-Wave 2X energy time few sec D-Wave Ising pulls out ALL non-zero Fourier coefficients (frequencies)
47 E. SUMMARY first demonstration of sparse coding using quantum computer! benchmark results on standard image classification task! mapping of visual features to D-Wave Chimera! compare D-Wave performance with GUROBI! compressive sensing on Ising for density profile detection where D-Wave significantly outperforms GUROBI!
48 E. FUTURE WORK optimize features add colors TrueNorth comparison hierarchy model compressive sensing on real images
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