A Tensor-Language Interface for Constrained Optimization

parameters// Input tensors (training data)
	    T 3     // The tensor we want to decompose
    variables // Variable tensors (model weights)
	    C 3     // The core of the decomposition
	    u 2, v 2, w 2 // One matrix for each axis
    min       // The objective function
	    norm(T - #(ijk,ai,bj,ck->abc;C,u,v,w))^2
    

parameters// Input tensors (training data)
	    X 4, y 1// Features X and labels y
    variables // Variable tensors (model weights)
	    th0 3, th1 3, th2 3 // Components of theta
    cse       // Reusable (sub)expressions
	    theta=#(zax,xby,ycz->abc;th0,th1,th2)
    min       // The objective function
	    sum((#(nabc,abc->n;X,theta)-y)^2)
    st        // Optimization constraints
	    theta >= 0
    

parameters // Input tensors (training data)
	    D 2, W 2 // Degree and weight matrix
    variables  // Variable tensors
	    V 2      // Target matrix
    min        // The objective function
	    #(ji,jk,ki->;V,D-W,V)  // tr(V^T*(D-W)*V)
    st         // Optimization constraints
	    #(ji,jk->ik;V,V) == delta(1) // V^T*V = I
    

parameters// Input tensors (training data)
	    D0 2, D1 2, D2 2 // One matrix per feature
    variables // Variable tensors (model weights)
	    th0 2, th1, th2  // Cores in the tensor train
    cse       // Common subexpressions
	    theta = #(ax,xby,yc->abc;th0,th1,th2)
    max       // The objective function
	    sum(log(#(abc,an,bn,cn->n;theta,D0,D1,D2)^2))
    st        // Constraints of the optimization
	    ||theta||^2 == 1