Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Abstract

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For $m \ge 2$, we present a $m \times 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m \times 2$ biquadratic forms, and shows that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 \times 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions.