# Siamese neural network

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Siamese neural network is an artificial neural network that use the same weights while working in tandem on two different input vectors to compute comparable output vectors.[1][2][3] Often one of the output vectors is precomputed, thus forming a baseline against which the other output vector is compared. This is similar to comparing fingerprints or more technical as a distance function for Locality-sensitive hashing.

It is possible to make a kind of structure that are functional similar to a siamese network, but still implement slightly different function. This is typically used for comparing similar instances in different type sets.

Uses of similarity measures where a siamese network might be used are such things as recognizing handwritten checks, automatic detection of faces in camera images, and matching queries with indexed documents. The perhaps most well-known application of siamese networks are face recognition, where known images of people are precomputed and compared to an image from a turnstile or similar. It is not obvious at first, but there are two slightly different problems. One is recognizing a person among a large number of other persons, that is the facial recognition problem. DeepFace is an example of such a system.[3] In its most extreme form this is recognizing a single person at a train station or airport. The other is face verification, that is to verify whether the photo in a pass is the same as the person claiming he or she is the same person. The siamese network might be the same, but the implementation can be quite different.

## Learning

Learning in siamese networks can be done with triplet loss or contrastive loss. For learning by triplet loss a baseline vector (anchor image) is compared against a positive vector (truthy image) and a negative vector (falsy image). The negative vector will force learning in the network, while the positive vector will act like a regularizer. For learning by contrastive loss there must be a weight decay to regularize the weights, or some similar operation like a normalization.

A distance metric for a loss function must have the following properties[4]

• Non-negativity: ${\displaystyle \delta (x,y)\geq 0}$
• Identity of Discernible: ${\displaystyle \delta (x,y)=0\iff x=y}$
• Symmetry: ${\displaystyle \delta (x,y)=\delta (y,x)}$
• Triangle inequality: ${\displaystyle \delta (x,z)\leq \delta (x,y)+\delta (y,z)}$

In particular, the triplet loss algorithm is often defined with squared Euclidean distance at its core.

### Predefined metrics, Euclidean distance metric

The common learning goal is to minimize a distance metric. This gives a loss function like

{\displaystyle {\begin{aligned}{\text{if}}\,i=j\,{\text{then}}&\,\|\operatorname {f} \left(x^{(i)}\right)-\operatorname {f} \left(x^{(j)}\right)\|\,{\text{is small}}\\{\text{otherwise}}&\,\|\operatorname {f} \left(x^{(i)}\right)-\operatorname {f} \left(x^{(j)}\right)\|\,{\text{is large}}\end{aligned}}}
${\displaystyle i,j}$ are indexes into a set of vectors
${\displaystyle \operatorname {f} (\cdot )}$function implemented by the siamese network

This is the most common case, but it is also a special case implementing an Euclidean distance metric.

On a matrix form the previous is often expressed as

${\displaystyle \operatorname {\delta } (\mathbf {x} ^{(i)},\mathbf {x} ^{(j)})\approx (\mathbf {x} ^{(i)}-\mathbf {x} ^{(j)})^{T}(\mathbf {x} ^{(i)}-\mathbf {x} ^{(j)})}$

This is not the same as it is the squared Euclidean distance, that is the Manhattan distance.

### Learned metrics, nonlinear distance metric

A more general case is where the output vector from the siamese network is passed through additional network layers implementing non-linear distance metrics.

{\displaystyle {\begin{aligned}{\text{if}}\,i=j\,{\text{then}}&\,\operatorname {\delta } \left[\operatorname {f} \left(x^{(i)}\right),\,\operatorname {f} \left(x^{(j)}\right)\right]\,{\text{is small}}\\{\text{otherwise}}&\,\operatorname {\delta } \left[\operatorname {f} \left(x^{(i)}\right),\,\operatorname {f} \left(x^{(j)}\right)\right]\,{\text{is large}}\end{aligned}}}
${\displaystyle i,j}$ are indexes into a set of vectors
${\displaystyle \operatorname {f} (\cdot )}$function implemented by the siamese network
${\displaystyle \operatorname {\delta } (\cdot )}$function implemented by the network joining outputs from the siamese network

On a matrix form the previous is often approximated as a Mahalanobis distance for a linear space as[5]

${\displaystyle \operatorname {\delta } (\mathbf {x} ^{(i)},\mathbf {x} ^{(j)})\approx (\mathbf {x} ^{(i)}-\mathbf {x} ^{(j)})^{T}\mathbf {M} (\mathbf {x} ^{(i)}-\mathbf {x} ^{(j)})}$

This can be further subdivided in at least Unsupervised learning and Supervised learning.

### Learned metrics, half-twin networks

This form also allows the siamese network to be more of a half-twin, implementing a slightly different functions

{\displaystyle {\begin{aligned}{\text{if}}\,i=j\,{\text{then}}&\,\operatorname {\delta } \left[\operatorname {f} \left(x^{(i)}\right),\,\operatorname {g} \left(x^{(j)}\right)\right]\,{\text{is small}}\\{\text{otherwise}}&\,\operatorname {\delta } \left[\operatorname {f} \left(x^{(i)}\right),\,\operatorname {g} \left(x^{(j)}\right)\right]\,{\text{is large}}\end{aligned}}}
${\displaystyle i,j}$ are indexes into a set of vectors
${\displaystyle \operatorname {f} (\cdot ),\operatorname {g} (\cdot )}$function implemented by the half-twin network
${\displaystyle \operatorname {\delta } (\cdot )}$function implemented by the network joining outputs from the siamese network

## References

1. ^ Bromley, Jane; Guyon, Isabelle; LeCun, Yann; Säckinger, Eduard; Shah, Roopak (1994). "Signature verification using a" siamese" time delay neural network" (PDF). Advances in neural information processing systems: 737–744.
2. ^ Chopra, S.; Hadsell, R.; LeCun, Y. (June 2005). "Learning a similarity metric discriminatively, with application to face verification". 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05). 1: 539–546 vol. 1. doi:10.1109/CVPR.2005.202.
3. ^ a b Taigman, Y.; Yang, M.; Ranzato, M.; Wolf, L. (June 2014). "DeepFace: Closing the Gap to Human-Level Performance in Face Verification". 2014 IEEE Conference on Computer Vision and Pattern Recognition: 1701–1708. doi:10.1109/CVPR.2014.220.
4. ^ Chatterjee, Moitreya; Luo, Yunan. "Similarity Learning with (or without) Convolutional Neural Network" (PDF). Retrieved 2018-12-07.
5. ^ Chandra, M.P. (1936). "On the generalized distance in statistics" (PDF). Proceedings of the National Institute of Sciences of India. 1. 2: 49–55.