A Note On the Use of Determinant for Proving Lower Bounds on the Size of Linear Circuits
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Abstract:
We consider computations of linear forms over R by circuits with linear gates where the absolute values coefficients are bounded by a constant. Also we consider a related concept of restricted rigidity of a matrix. We prove some lower bounds on the size of such circuits and the restricted rigidity of matrices in terms of the absolute value of the determinant of the matrix. 1 The purpose of this note is to analyze the role of the determinant in establishing lower bounds for linear circuits over R with bounds on the coefficients. The study of the linear complexity of computation has a long history, starting with the seminal works of Morgenstern [6, 7], Grigoriev [4] and Valiant [10]. But it is still an open problem to prove more than linear lower bounds on general circuits computing an explicitly defined linear form. With the restriction on the size of coefficients, Morgenstern [7] proved nontrivial lower
Citations
| 104 | Algebraic Complexity Theory – Bürgisser, Clausen, et al. - 1997 |
| 65 | Graph–theoretic arguments in low–level complexity – Valiant - 1977 |
| 29 | Spectral methods for matrix rigidity with applications to size-depth trade-offs and communication complexity – Lokam |
| 16 | Note on a lower bound of the linear complexity of the fast Fourier transform – Morgenstern - 1973 |
| 13 | Introduction to Matrix Analysis, 2nd ed – Bellman - 1970 |
| 12 | Razborov Improved Lower Bounds on the Rigidity of Hadamard Matrices – Kashin, A - 1997 |
| 8 | A Spectral Approach to Lower Bounds – Chazelle - 1994 |
| 7 | On the complexity of bilinear forms – Nisan, Wigderson - 1995 |
| 3 | Using the notion of separability and independence for proving lower bounds on circuit complexity – Grigoriev - 1976 |
| 2 | The Linear Complexity of – Morgenstern - 1975 |
