This document discusses using tensor decomposition models for context-aware recommendation systems. It introduces tensor ring decomposition as a new model and describes its advantages over other tensor decomposition models like CP and Tucker in handling high-order data without dimensionality issues. The document proposes using a max-norm regularization with tensor ring decomposition to build recommendation models from tensorized contextual data like tags and timestamps, experimenting on MovieLens datasets.

1. Research on context-aware recommendation system
based on tensor decomposition
by Longhao Yuan
Feb 2020
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 1 / 9

2. What is tensor?
Tensors are multi-dimensional arrays.
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 2 / 9

3. Benefit of tensor
Retain structure information of high-order data.
Able to present high-order relation of data.
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 3 / 9

4. Traditional tensor decomposition models
CP, Tucker and TT models.
Tucker and CP are the most common-used models in RS field.
Drawback of Tucker: curse of dimensionality (IN)
Drawback of CP: difficult to optimize.
Drawback of TT: strict rank constraint.
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 4 / 9

5. Tensor ring decomposition (TRD)
Model introduction
Advantages and potential
High compressive ability and free from the curse of
dimensionality(NIR2
).
Suitable for large-scale and high-order data.
State-of-the-art results in signal processing and image recovery.
Currently, there is no study about TR model application in RS field.
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 5 / 9

6. Solve RS problem based on tensor decomposition
Consider RS problem as a tensor completion model.
Challenges: large-scale → high computational cost, high sparsity →
overfitting
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 6 / 9

7. General models and algorithms for RS
Explicit feedback
Lexplicit =
n∈M+
(ym − xm)2
+ λ
N
n=1
G(n) 2
F , s.t. xm = ψm([G]) (1)
where M+ is all the observed interaction, ym and xm are the mth real
interaction and tensor decomposition approximation respectively.
Implicit feedback
Limplicit =
n∈M
wm(ym − xm)2
+ λ
N
n=1
G(n) 2
F , s.t. xm = ψm([G]) (2)
where M is all the interactions which considers the unobserved entries (0)
as negative interaction, wm is the weight of each entry calculated by
wm = 1 + γlog(1 + ym).
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 7 / 9

8. My algorithm: Max-norm regularized TRD (unpublished)
Background
Matrix max-norm is an efficient low-rank regularizer.
My contribution
Extend it to tensor based on TR model and propose theoretical supports.
Model and algorithm
min
X=Ψ([G])
f (X, T ; M), s.t. X TR−max ≤ λ (3)
solved by projected SGD (mini-batch).
Advantages: robust to overfitting and rank selection
Experiments: hyperspectral image cloud removal, RS data fitting
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 8 / 9

9. Experiment on MoiveLens dataset
data pre-processing: k-core method
Trim the data entries, if any of user, movie or tag appears less than k
times.
implicit data formulation
Structure: user × movie × tag × month → tag 0 − 1)
Size: 828 × 3125 × 2539 × 36 (volume 5.4e11) of sparsity 1.5e − 7 by
core-3 trim.
Explicit data formulation
Structure: (user × movie × month → ratings 0-5)
Size: 2045 × 3696 × 36 (volume 2.7e8) of sparsity 1.6e − 3 by core-20
trim.
Code is available at
https://github.com/yuanlonghao/Tensor-based-recommendation-system
by Longhao Yuan Research on context-aware recommendation system based on tensor decompositionFeb 2020 9 / 9