t-SNE is often used as a better alternative than PCA in terms of visualization. This paper is the one that proposed it. It is an extension to SNE (stochastic neighbor embedding), which the first few page of the paper outlined it:

## SNE

Assume we have points $$x_1, \cdots, x_N$$ in high dimensional coordinates. The similarity of points $$x_i$$ and $$x_j$$ is defined as

$p_{j\mid i} = \frac{\exp(-\Vert x_i - x_j\Vert^2 / 2\sigma_i^2)}{\sum_{k\ne i}\exp(-\Vert x_i - x_k\Vert^2 / 2\sigma_i^2)}$

where the $$\sigma_i^2$$ is the scale parameter to the Gaussian density function, and it is specific to $$x_i$$. The similarity of a point to itself is defined to be zero, i.e., $$p_{i\mid i}=0$$.

If we have a low-dimensional mapping $$y_i$$ for each point $$x_i$$, we can do the same to calculate similarity $$q_{j\mid i}$$ (but the paper suggest to use a constant $$\sigma_i^2=\frac12$$ to remove the denominators in exponential functions, to simplify the case in low dimension).

The idea of SNE is that, if the mapping from $$x_i$$ to $$y_i$$ are perfect, then $$p_{j\mid i}=q_{j\mid i}$$ for all $$i,j$$ and we can use the Kullback-Leiber divergence as the measure of mismatch. We should find $$y_i$$ to minimize the sum of all K-L divergence, i.e. use the following as cost function:

$C = \sum_i KL(P_i\Vert Q_i) = \sum_i\sum_j p_{j\mid i}\log\frac{p_{j\mid i}}{q_{j\mid i}}$

we can seek for the minimizer $$y_i$$ using gradient descent, where

$\frac{\partial C}{\partial y_i} = 2\sum_j (p_{j\mid i} - q_{j\mid i} + p_{i\mid j} - q_{i\mid j})(y_i - y_j)$

The variance in $$p_{j\mid i}$$ is set such that points $$x_i$$ at denser regions would use smaller $$\sigma_i^2$$, as Shannon entropy $$H$$ increases with $$\sigma_i^2$$:

$H(P_i) = -\sum_k p_{j\mid i}\log_2 p_{j\mid i}$

Hence we define perplexity as $$2^{H(P_i)}$$, which is a measure of effective number of neighbors for point $$x_i$$. A typical value of perplexity, as suggested by the paper, is between 5 and 50. The variance $$\sigma_i^2$$ should be set such that the perplexity are equal for all the points. This can be found by binary search (seems like the perplexity is a monotonic function of variance).

From the partial derivative $$\partial C/\partial y_i$$ we can see that it is a force in the direction of $$y_i - y_j$$, i.e., point $$y_i$$ are subject to forces toward each other points $$y_j$$. The magnitude of the force is proportional to the stiffness $$(p_{j\mid i} - q_{j\mid i} + p_{i\mid j} - q_{i\mid j})$$, which $$p_{j\mid i}+p_{i\mid j}$$ is positive and $$q_{j\mid i}+q_{i\mid j}$$ is negative. This means if the similarity is too close in lower dimension $$y_i$$ compared to higher dimension $$x_i$$, there will be a net force to push the point away, and vice versa. Equalibrium will be attained when the attraction and repulsion magnitudes are equal.

The paper suggested to use gradient descent with momentum, i.e.,

$y^{(t)} = y^{(t-1)} + \eta \frac{\partial C}{\partial y} + \alpha(t)\big(y^{(t-1)} - y^{(t-2)}\big)$

## t-SNE

The SNE above is using Gaussian kernels, as $$p_{j\mid i}$$ and $$q_{j\mid i}$$ are both using Gaussian density function. The t-SNE is to use Cauchy distribution (i.e., t distribution with dof=1) for $$q_{j\mid i}$$, and normalized over all pairs $$y_i,y_j$$:

$q_{ij} = \frac{(1+\Vert y_i - y_j\Vert^2)^{-1}}{\sum_{k\ne\ell}(1+\Vert y_k - y_{\ell}\Vert^2)^{-1}}$

and the similarity in high dimension is symmetrizied:

$p_{ij} = \frac{p_{i\mid j}+p_{j\mid i}}{2}$

and normalized over the row (i.e. $$p_{ij}$$ normalized across all $$j$$).

With the cost function defined in the same way, the gradient is now:

\begin{aligned} C &= KL(P\Vert Q) = \sum_i \sum_j p_{ij} \log\frac{p_{ij}}{q_{ij}} \\ \frac{\partial C}{\partial y_i} &= 4\sum_j (p_{ij} - q_{ij})(y_i - y_j)(1+\Vert y_i - y_j\Vert^2)^{-1} \end{aligned}

The reason for the t-distribution for $$q_{ij}$$ and Gaussian for $$p_{ij}$$ is that t-distribution is heavytailed and there is a unique sweet spot of separation. Therefore, it will maintain a reasonable scale in low dimension for $$q_{ij}$$.

## Implementation

The reference implementation is from the author: https://github.com/lvdmaaten/bhtsne/blob/master/tsne.cpp and below is how I ported it into Python.

import datetime
import sys
import numpy as np

def tSNE(X, no_dims=2, perplexity=30, seed=0, max_iter=1000, stop_lying_iter=100, mom_switch_iter=900):
"""The t-SNE algorithm

Args:
X: the high-dimensional coordinates
no_dims: number of dimensions in output domain
Returns:
Points of X in low dimension
"""
momentum = 0.5
final_momentum = 0.8
eta = 200.0
N, _D = X.shape
np.random.seed(seed)

# normalize input
X -= X.mean(axis=0) # zero mean
X /= np.abs(X).max() # min-max scaled

# compute input similarity for exact t-SNE
P = computeGaussianPerplexity(X, perplexity)
# symmetrize and normalize input similarities
P = P + P.T
P /= P.sum()
# lie about the P-values
P *= 12.0
# initialize solution
Y = np.random.randn(N, no_dims) * 0.0001
# perform main training loop
gains = np.ones_like(Y)
uY = np.zeros_like(Y)
for i in range(max_iter):
# compute gradient, update gains
dY = computeExactGradient(P, Y)
gains = np.where(np.sign(dY) != np.sign(uY), gains+0.2, gains*0.8).clip(0.1)
# gradient update with momentum and gains
uY = momentum * uY - eta * gains * dY
Y = Y + uY
# make the solution zero-mean
Y -= Y.mean(axis=0)
# Stop lying about the P-values after a while, and switch momentum
if i == stop_lying_iter:
P /= 12.0
if i == mom_switch_iter:
momentum = final_momentum
# print progress
if (i % 50) == 0:
C = evaluateError(P, Y)
now = datetime.datetime.now()
print(f"{now} - Iteration {i}: Error = {C}")
return Y

"""Gradient of t-SNE cost function

Args:
P: similarity matrix
Y: low-dimensional coordinates
Returns:
dY, a numpy array of shape (N,D)
"""
N, _D = Y.shape
# compute squared Euclidean distance matrix of Y, the Q matrix, and the normalization sum
DD = computeSquaredEuclideanDistance(Y)
Q = 1/(1+DD)
sum_Q = Q.sum()
mult = (P - (Q/sum_Q)) * Q
dY = np.zeros_like(Y)
for n in range(N):
for m in range(N):
if n==m: continue
dY[n] += (Y[n] - Y[m]) * mult[n,m]
return dY

def evaluateError(P, Y):
"""Evaluate t-SNE cost function

Args:
P: similarity matrix
Y: low-dimensional coordinates
Returns:
Total t-SNE error C
"""
DD = computeSquaredEuclideanDistance(Y)
# Compute Q-matrix and normalization sum
Q = 1/(1+DD)
np.fill_diagonal(Q, sys.float_info.min)
Q /= Q.sum()
# Sum t-SNE error: sum P log(P/Q)
error = P * np.log( (P + sys.float_info.min) / (Q + sys.float_info.min) )
return error.sum()

def computeGaussianPerplexity(X, perplexity):
"""Compute Gaussian Perplexity

Args:
X: numpy array of shape (N,D)
perplexity: double
Returns:
Similarity matrix P
"""
# Compute the squared Euclidean distance matrix
N, _D = X.shape
DD = computeSquaredEuclideanDistance(X)
# Compute the Gaussian kernel row by row
P = np.zeros_like(DD)
for n in range(N):
found = False
beta = 1.0
min_beta = -np.inf
max_beta = np.inf
tol = 1e-5

# iterate until we get a good perplexity
n_iter = 0
while not found and n_iter < 200:
# compute Gaussian kernel row
P[n] = np.exp(-beta * DD[n])
P[n,n] = sys.float_info.min
# compute entropy of current row
# Gaussians to be row-normalized to make it a probability
# then H = sum_i -P[i] log(P[i])
#        = sum_i -P[i] (-beta * DD[n] - log(sum_P))
#        = sum_i P[i] * beta * DD[n] + log(sum_P)
sum_P = P[n].sum()
H = beta * (DD[n] @ P[n]) / sum_P + np.log(sum_P)
# Evaluate if entropy within tolerance level
Hdiff = H - np.log2(perplexity)
if -tol < Hdiff < tol:
found = True
break
if Hdiff > 0:
min_beta = beta
if max_beta in (np.inf, -np.inf):
beta *= 2
else:
beta = (beta + max_beta) / 2
else:
max_beta = beta
if min_beta in (np.inf, -np.inf):
beta /= 2
else:
beta = (beta + min_beta) / 2
n_iter += 1
# normalize this row
P[n] /= P[n].sum()
assert not np.isnan(P).any()
return P

def computeSquaredEuclideanDistance(X):
"""Compute squared distance
Args:
X: numpy array of shape (N,D)
Returns:
numpy array of shape (N,N) of squared distances
"""
N, _D = X.shape
DD = np.zeros((N,N))
for i in range(N-1):
for j in range(i+1, N):
diff = X[i] - X[j]
DD[j][i] = DD[i][j] = diff @ diff
return DD

def main():
import tensorflow as tf
(X_train, y_train), (X_test, y_test) = tf.keras.datasets.mnist.load_data()
print("Dimension of X_train:", X_train.shape)
print("Dimension of y_train:", y_train.shape)
print("Dimension of X_test:", X_test.shape)
print("Dimension of y_test:", y_test.shape)
n = 200
rows = np.random.choice(X_train.shape, n, replace=False)
X_data = X_train[rows].reshape(n, -1).astype("float")
X_label = y_train[rows]
Y = tSNE(X_data, 2, 30, 0, 1000, 100, 900)
np.savez("data.npz", X=X_data, Y=Y, label=X_label)

if __name__ == "__main__":
main()


From the code, we see a few tweaks that are not mentioned in the paper:

In tSNE(), the main algorithm:

• The initial points of $$y_i$$ are randomized using $$N(\mu=0,\sigma=10^{-4})$$. The small standard deviation will make the initial points closely packed.
• Gradient descent used in the algorithm is not exactly like that shown in the formula above. There is a gain factor multiplied to the gradient descent. It started as 1 and updated according to the sign of $$\partial C/\partial y_i$$ and $$(y_i^{(t-1)} - y_i^{(t-2)})$$. It will increase by 0.2 if signs differ and decrease to its 80% if signs agree, with the lowerbound of 0.1; this is to make the gradient descent do larger steps if we are moving in the right direction
• Initially the values of $$p_{ij}$$ are multiplied by 12 and this amplified version of $$P$$ matrix is used to calculate the gradient in the beginning, then scaled back to the original values of $$P$$
• The momentum used in gradient descent will be switched from 0.5 to 0.8 at later stage

In computeExactGradient()

• We first compute the multiplier $$(p_{ij}-q_{ij})(1+\Vert y_i-y_j\Vert^2)^{-1}$$ as matrix mult to make it computationally more efficient
• While the gradient formula above carried a coefficient 4, it is not used. Instead, we use a bigger (200) $$\eta$$ in tSNE() to compensate

In computeGaussianPerplexity():

• The binary search is using a tolerance $$10^{-5}$$, which the entropy calculated would have at most this much of error from the log of the target perplexity
• Instead of doing bisection search on $$\sigma_i^2$$, it is searching on $$\beta=1/2\sigma_i^2$$. Hence $$\beta$$ is larger for a smaller entropy $$H$$. The search will reduce $$\beta$$ for overshot $$H$$.
• The matrix $$P$$ is normalized by row, but the diagonal entries are using sys.float_info.min which is the smallest positive float in the system. This is essentially zero but avoids division-by-zero error in corner cases.

## Remark:

The PDF of t-distribution with dof $$\nu>0$$ is defined for $$x\in(-\infty,\infty)$$:

$f(x) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}$

which $$\nu=1$$ gives Cauchy distribution with location $$x_0=0$$ and scale $$\gamma=1$$:

\begin{aligned} f(x) &= \frac{\Gamma(1)}{\sqrt{\pi}\Gamma(\frac12)} (1+x^2)^{-1} \\ &= \frac{1}{\sqrt{\pi}\sqrt{\pi}} (1+x^2)^{-1} \\ &= \frac{1}{\pi (1+x^2)} \end{aligned}

## Bibliographic data

@article{
title = "Visualizing Data using t-SNE",
author = "Lauren van der Maaten and Geoffrey Hinton",
journal = "Journal of Machine Learning Research",
volume = "9",
number = "86",
year = "2008",
pages = "2579--2605",
}