Quantum Recommendation Systems

Transcription

1 Quantum Recommendation Systems Iordanis Kerenidis 1 Anupam Prakash 2 1 CNRS, Université Paris Diderot, Paris, France, EU. 2 Nanyang Technological University, Singapore. April 4, 2017

2 The HHL algorithm Utilize intrinsic linear algebra capabilities of quantum computers for exponential speedups.

3 The HHL algorithm Utilize intrinsic linear algebra capabilities of quantum computers for exponential speedups. Vector state x = i x i i where x R n is a unit vector.

4 The HHL algorithm Utilize intrinsic linear algebra capabilities of quantum computers for exponential speedups. Vector state x = i x i i where x R n is a unit vector. Given sparse matrix A R n n and b there is a quantum algorithm to prepare A 1 b in time polylog(n). [Harrow, Hassidim, Lloyd]

5 The HHL algorithm Utilize intrinsic linear algebra capabilities of quantum computers for exponential speedups. Vector state x = i x i i where x R n is a unit vector. Given sparse matrix A R n n and b there is a quantum algorithm to prepare A 1 b in time polylog(n). [Harrow, Hassidim, Lloyd] Assumptions: b can be prepared polylog(n) time and A is polylog(n) sparse.

6 The HHL algorithm Utilize intrinsic linear algebra capabilities of quantum computers for exponential speedups. Vector state x = i x i i where x R n is a unit vector. Given sparse matrix A R n n and b there is a quantum algorithm to prepare A 1 b in time polylog(n). [Harrow, Hassidim, Lloyd] Assumptions: b can be prepared polylog(n) time and A is polylog(n) sparse. Incomparable to classical linear system solver which returns vector x R n as opposed to x.

7 Quantum Machine Learning HHL led to several proposals for quantum machine learning algorithms.

8 Quantum Machine Learning HHL led to several proposals for quantum machine learning algorithms. Principal components analysis, classification with l 2 -SVMs, k-means clustering, perceptron, nearest neighbors... [Lloyd, Mohseni, Rebentrost, Wiebe, Kapoor, Svore]

9 Quantum Machine Learning HHL led to several proposals for quantum machine learning algorithms. Principal components analysis, classification with l 2 -SVMs, k-means clustering, perceptron, nearest neighbors... [Lloyd, Mohseni, Rebentrost, Wiebe, Kapoor, Svore] Algorithms achieve exponential speedups only for sparse/well-conditioned data.

10 Quantum Machine Learning HHL led to several proposals for quantum machine learning algorithms. Principal components analysis, classification with l 2 -SVMs, k-means clustering, perceptron, nearest neighbors... [Lloyd, Mohseni, Rebentrost, Wiebe, Kapoor, Svore] Algorithms achieve exponential speedups only for sparse/well-conditioned data. Sometimes a variant of the classical problem is solved: l 1 vs l 2 -SVM.

11 Quantum Machine Learning HHL led to several proposals for quantum machine learning algorithms. Principal components analysis, classification with l 2 -SVMs, k-means clustering, perceptron, nearest neighbors... [Lloyd, Mohseni, Rebentrost, Wiebe, Kapoor, Svore] Algorithms achieve exponential speedups only for sparse/well-conditioned data. Sometimes a variant of the classical problem is solved: l 1 vs l 2 -SVM. Incomparable with classical.

12 Quantum Recommendation Systems Open problem: A quantum machine learning algorithm with exponential worst case speedup for classical problem.

13 Quantum Recommendation Systems Open problem: A quantum machine learning algorithm with exponential worst case speedup for classical problem. Quantum recommendation systems.

14 Quantum Recommendation Systems Open problem: A quantum machine learning algorithm with exponential worst case speedup for classical problem. Quantum recommendation systems. An exponential speedup over classical with similar assumptions and guarantees.

15 Quantum Recommendation Systems Open problem: A quantum machine learning algorithm with exponential worst case speedup for classical problem. Quantum recommendation systems. An exponential speedup over classical with similar assumptions and guarantees. An end to end application with no assumptions on the data set.

16 Quantum Recommendation Systems Open problem: A quantum machine learning algorithm with exponential worst case speedup for classical problem. Quantum recommendation systems. An exponential speedup over classical with similar assumptions and guarantees. An end to end application with no assumptions on the data set. Solves the same problem as a classical recommendation system.

17 The Recommendation Problem The preference matrix P. P 1 P 2 P 3 P 4 P n 1 P n U 1 U 2 U 3. U m.1.4???.9.2?.6?.85???.8.9?.2?.75???.2

18 The Recommendation Problem The preference matrix P. P 1 P 2 P 3 P 4 P n 1 P n U 1 U 2 U 3. U m.1.4???.9.2?.6?.85???.8.9?.2?.75???.2 P ij is the value of item j for user i. Samples from P arrive in an online manner.

19 The Recommendation Problem The preference matrix P. P 1 P 2 P 3 P 4 P n 1 P n U 1 U 2 U 3. U m.1.4???.9.2?.6?.85???.8.9?.2?.75???.2 P ij is the value of item j for user i. Samples from P arrive in an online manner. The assumption that P has a good rank-k approximation for small k is widely used.

20 The Netflix problem

21 Reconstruction vs sampling Matrix reconstruction algorithms reconstruct P P using the low rank assumption and require time poly(mn).

22 Reconstruction vs sampling Matrix reconstruction algorithms reconstruct P P using the low rank assumption and require time poly(mn). A reconstruction based recommendation system requires time poly(n), even with pre-computation.

23 Reconstruction vs sampling Matrix reconstruction algorithms reconstruct P P using the low rank assumption and require time poly(mn). A reconstruction based recommendation system requires time poly(n), even with pre-computation. Matrix sampling suffices to obtain good recommendations.

24 Reconstruction vs sampling Matrix reconstruction algorithms reconstruct P P using the low rank assumption and require time poly(mn). A reconstruction based recommendation system requires time poly(n), even with pre-computation. Matrix sampling suffices to obtain good recommendations. Quantum algorithms can perform matrix sampling.

25 Reconstruction vs sampling Matrix reconstruction algorithms reconstruct P P using the low rank assumption and require time poly(mn). A reconstruction based recommendation system requires time poly(n), even with pre-computation. Matrix sampling suffices to obtain good recommendations. Quantum algorithms can perform matrix sampling. Theorem There is a quantum recommendation algorithm with running time O(poly(k)polylog(mn)).

26 Computational Model Samples from P arrive in an online manner and are stored in data structure with update time O(log 2 mn).

27 Computational Model Samples from P arrive in an online manner and are stored in data structure with update time O(log 2 mn). The quantum algorithm has oracle access to binary tree data structure storing additional metadata

28 Computational Model Samples from P arrive in an online manner and are stored in data structure with update time O(log 2 mn). The quantum algorithm has oracle access to binary tree data structure storing additional metadata We use the standard memory model used for algorithms like Grover search.

29 Computational Model Samples from P arrive in an online manner and are stored in data structure with update time O(log 2 mn). The quantum algorithm has oracle access to binary tree data structure storing additional metadata We use the standard memory model used for algorithms like Grover search. Users arrive into system in an online manner and system provides recommendations in time poly(k)polylog(mn).

30 Singular value estimation The singular value decomposition for matrix A is written as A = i σ iu i v t i.

31 Singular value estimation The singular value decomposition for matrix A is written as A = i σ iu i vi t. The rank-k approximation A k = i [k] σ iu i vi t minimizes A A k F.

32 Singular value estimation The singular value decomposition for matrix A is written as A = i σ iu i vi t. The rank-k approximation A k = i [k] σ iu i vi t minimizes A A k F. Quantum singular value estimation:

33 Singular value estimation The singular value decomposition for matrix A is written as A = i σ iu i vi t. The rank-k approximation A k = i [k] σ iu i vi t minimizes A A k F. Quantum singular value estimation: Theorem There is an algorithm with running time O(polylog(mn)/ɛ) that transforms i α i v i i α i v i σ i where σ i σ i ± ɛ A F with probability at least 1 1/poly(n).

34 Matrix Sampling Let T be a 0/1 matrix such that T ij = 1 if item j is good recommendation for user i. P 1 P 2 P 3 P 4 P n 1 P n U 1 U 2 U 3. U m 0 0??? 1 0? 0? 1??? 1 1? 0? 1??? 0

35 Matrix Sampling Let T be a 0/1 matrix such that T ij = 1 if item j is good recommendation for user i. P 1 P 2 P 3 P 4 P n 1 P n U 1 U 2 U 3. U m 0 0??? 1 0? 0? 1??? 1 1? 0? 1??? 0 Set the?s to 0 and rescale to obtain a subsample matrix T.

36 Matrix Sampling P Rounding T Uniform subsample T Low rank assumption SVD, QSVE T k [Achlioptas, McSherry] extended T k Figure: Matrix sampling based recommendation system.

37 Matrix Sampling T is the binary recommendation matrix obtained by rounding P.

38 Matrix Sampling T is the binary recommendation matrix obtained by rounding P. T is a uniform subsample of T : { A ij /p [with probability p] Â ij = 0 [otherwise]

39 Matrix Sampling T is the binary recommendation matrix obtained by rounding P. T is a uniform subsample of T : { A ij /p [with probability p] Â ij = 0 [otherwise] T k and T k are rank-k approximations for T and T.

40 Matrix Sampling T is the binary recommendation matrix obtained by rounding P. T is a uniform subsample of T : { A ij /p [with probability p] Â ij = 0 [otherwise] T k and T k are rank-k approximations for T and T. The low rank assumption implies that T T k ɛ T F for small k.

41 Matrix Sampling T is the binary recommendation matrix obtained by rounding P. T is a uniform subsample of T : { A ij /p [with probability p] Â ij = 0 [otherwise] T k and T k are rank-k approximations for T and T. The low rank assumption implies that T T k ɛ T F for small k. Analysis: Sampling from matrix close to T k yields good recommendations.

42 Analysis Samples from T k are good recommendations, for large fraction of typical users.

43 Analysis Samples from T k are good recommendations, for large fraction of typical users. Sampling from T k suffices.

44 Analysis Samples from T k are good recommendations, for large fraction of typical users. Sampling from T k suffices. Theorem (AM02) If  is obtained from a 0/1 matrix A by subsampling with probability p = 16n/η A 2 F then with probability at least 1 exp( 19(log n) 4 ), for all k, A Âk F A A k F + 3 ηk 1/4 A F

45 Analysis The quantum algorithm samples from T σ,κ, a projection onto all singular values σ and some in the range [(1 κ)σ, σ).

46 Analysis The quantum algorithm samples from T σ,κ, a projection onto all singular values σ and some in the range [(1 κ)σ, σ). We extend AM02 to this setting showing that: T T σ,κ F 9ɛ T F

47 Analysis The quantum algorithm samples from T σ,κ, a projection onto all singular values σ and some in the range [(1 κ)σ, σ). We extend AM02 to this setting showing that: T T σ,κ F 9ɛ T F For most typical users, samples from ( T σ,κ ) i are good recommendations with high probability.

48 Quantum Recommendation Algorithm Prepare state T i corresponding to row for user i.

49 Quantum Recommendation Algorithm Prepare state T i corresponding to row for user i. Apply quantum projection algorithm to T i to obtain ( T σ,κ ) i.

50 Quantum Recommendation Algorithm Prepare state T i corresponding to row for user i. Apply quantum projection algorithm to T i to obtain ( T σ,κ ) i. Measure projected state in computational basis to get recommendation.

51 Quantum Recommendation Algorithm Prepare state T i corresponding to row for user i. Apply quantum projection algorithm to T i to obtain ( T σ,κ ) i. Measure projected state in computational basis to get recommendation. The threshold σ = ɛ p A 2k F and κ = 1 3.

52 Quantum Recommendation Algorithm Prepare state T i corresponding to row for user i. Apply quantum projection algorithm to T i to obtain ( T σ,κ ) i. Measure projected state in computational basis to get recommendation. The threshold σ = ɛ p A 2k F and κ = 1 3. Running time depends on the threshold and not the condition number.

53 The projection algorithm Let A = i σ iu i vi t be the singular value decomposition, write input x = i α i v i.

54 The projection algorithm Let A = i σ iu i vi t be the singular value decomposition, write input x = i α i v i. Estimate singular values i α i v i σ i to additive error κσ/2.

55 The projection algorithm Let A = i σ iu i vi t be the singular value decomposition, write input x = i α i v i. Estimate singular values i α i v i σ i to additive error κσ/2. Map to i α i v i σ i t where t = 1 if σ i (1 κ/2)σ and erase σ i.

56 The projection algorithm Let A = i σ iu i vi t be the singular value decomposition, write input x = i α i v i. Estimate singular values i α i v i σ i to additive error κσ/2. Map to i α i v i σ i t where t = 1 if σ i (1 κ/2)σ and erase σ i. Post-select on t = 1.

57 The projection algorithm Let A = i σ iu i vi t be the singular value decomposition, write input x = i α i v i. Estimate singular values i α i v i σ i to additive error κσ/2. Map to i α i v i σ i t where t = 1 if σ i (1 κ/2)σ and erase σ i. Post-select on t = 1. The output A σ,κ x a projection the space of singular vectors with singular values σ and some in the range [(1 κ)σ, σ).

58 Open Questions Find a classical algorithm matrix sampling based recommendation algorithm that runs in time O(poly(k)polylog(mn)). OR Prove a lower bound to rule out such an algorithm. Find more quantum machine learning algorithms.