SGH-hashing

This passage adopts a sequential learning strategy in a bit-wise manner.

Problem Definition
datapoints $X=[x_1;x_2;...;x_n]^\mathrm{T}\in\mathscr R^{n\times d}$,where d is the dimensionality of the data points.
We use c binary hash functions $\{h_k(\cdot)\mid k=1,2,...,c\}$ to compute the binary code of $x_i$, i.e., $b_i=\lbrack h_1(x_i),h_2(x_i),...,h_c(x_i)\rbrack^\mathrm{T}$
$\mathrm S_{ij}=e^-\frac{{\parallel x_i-x_j\parallel^2}_F }{\rho}$,donote the similarity in the original space. where $\rho>0$
Objective function

$$min\sum_{i,j=1}^n(\widetilde S_{ij}-\frac{1}{c}b_i^\mathrm{T}b_j)^2$$ where $\widetilde S_{ij}=2S_{ij}-1$
According to KSH we define the hash function for the k-th bit of $b_i$ as follows:
$$h_k(x_i)=sgn(\sum_{j=1}^mW_{kj}\phi(x_i,x_j)+bias_k)$$
where $W\in\mathscr R^{c\times m}$ is the weight matrix ,$\phi(x_i,x_j)$is a kernel function, m denotes the number of kernel bases. In fact, the above function can be written as $h_k(\mathbf{x})=sgn(K({x}){w_k})$ where $w_k=W_{k*}^\mathrm{T}$ where $W_{k*}$ reprensents the k row of $W$ ,$K(x) \in\mathscr R^{1\times m}$.
So we can get the object function with the parameter $W$ as :
$$min\parallel c\widetilde S-sgn(K(X)W^\mathrm{T})sgn(K(X)W^\mathrm{T})^\mathrm{T}\parallel$$