Efficient distance metric learning for fine-grained visual categorization
Methods and systems for distance metric learning include generating two random projection matrices of a dataset from a d-dimensional space into an m-dimensional sub-space, where m is smaller than d. An optimization problem is solved in the m-dimensional subspace to learn a distance metric based on the random projection matrices. The distance metric is recovered in the d-dimensional space.
1. A method for distance metric learning, comprising:
generating two random projection matrices of a dataset from a d-dimensional space into an m-dimensional sub-space, where m is smaller than d;
solving an optimization problem in the m-dimensional subspace to learn a distance metric based on the random projection matrices;
recovering the distance metric in the d-dimensional space by obtaining a low-rank approximation using a randomized approach to avoid storing the entire metric in the original space; and
classifying images using the recovered distance metric.
2. The method of claim 1 , wherein said steps of generating, solving, and recovering are performed iteratively.
3. The method of claim 2 , wherein triplets of only one epoch are selected in each iteration.
4. The method of claim 1 , wherein solving the optimization problem comprises performing stochastic gradient descent in the m-dimensional subspace.
5. The method of claim 1 , wherein solving the optimization problem comprises computing dual variables in the m-dimensional subspace.
6. The method of claim 1 , wherein the optimization problem is:
min
M
^
∈
S
m
λ
2
M
^
F
2
+
1
N
∑
t
=
1
N
l
(
〈
A
^
t
,
M
^
〉
)
where S m is the m-dimensional subspace, {circumflex over (M)} is a metric in the m-dimensional subspace, ∥·∥ F 2 is the Frobenius norm, λ is a weight for regularizer ∥{circumflex over (M)}∥ F 2 , l(.) is any smoothed convex loss function, is the dot product between two matrices, N is a number of triplets, and  t is a t th dual variable.
7. A system for distance metric learning, comprising:
a projection module configured to generate two random projection matrices of a dataset from a d-dimensional space into an m-dimensional sub-space, where m is smaller than d;
a processor configured to solve an optimization problem in the m-dimensional subspace to learn a distance metric based on the random projection matrices; and
a metric recovery module configured to recover the distance metric in the d-dimensional space by obtaining a low-rank approximation using a randomized approach to avoid storing the entire metric in the original space, wherein the processor is further configured to classify images using the recovered distance metric.
8. The system of claim 7 , wherein the projection module, processor, and metric recovery module are configured to operate iteratively.
9. The system of claim 8 , wherein triplets of only one epoch are selected in each iteration.
10. The system of claim 7 , wherein the processor is configured to solve the optimization problem using a stochastic gradient descent in the m-dimensional subspace.
11. The system of claim 7 , wherein the processor is configured to solve the optimization problem by computing dual variables in the m-dimensional subspace.
12. The system of claim 7 , wherein the optimization problem is:
min
M
^
∈
S
m
λ
2
M
^
F
2
+
1
N
∑
t
=
1
N
l
(
〈
A
^
t
,
M
^
〉
)
where S m is the m-dimensional subspace, {circumflex over (M)} is a metric in the m-dimensional subspace, ∥·∥ F 2 is the Frobenius norm, λ is a weight for regularizer ∥{circumflex over (M)}∥ F 2 , l(.) is any smoothed convex loss function, is the dot product between two matrices, N is a number of triplets, and  t is a t th dual variable.