Download Complexity Lower Bounds using Linear Algebra (Foundations by Satyanarayana V. Lokam PDF

By Satyanarayana V. Lokam

Show description

Read or Download Complexity Lower Bounds using Linear Algebra (Foundations and Trends in Theoretical Computer Science) PDF

Similar computers books

Real world Camera Raw with Adobe Photoshop CS: industrial strength production techniques

Name it a regulate factor, yet until eventually lately - or, extra specifically,until the supply of electronic uncooked digicam codecs - you simplyweren't able to make the movement to electronic images. uncooked formats,however, replaced all of that via permitting you to retrieve imagesbefore any in-camera processing has been played.

Information Networking. Convergence in Broadband and Mobile Networking: International Conference, ICOIN 2005, Jeju Island, Korea, January 31- February 2, 2005. Proceedings

Welcome to ICOIN 2005,the overseas convention on details Netwo- ing, held at Ramada Plaza Jeju inn, Jeju Island, Korea in the course of January 31– February2,2005. ICOIN2005followedthesuccessofpreviousconferences. considering the fact that 1986, the convention has supplied a technical discussion board for varied matters in inf- mation networking.

Simulated Evolution and Learning: First Asia-Pacific Conference, SEAL'96 Taejon, Korea, November 9–12, 1996 Seclected Papers

This publication constitutes the completely refereed post-conference documentation of the 1st Asia-Pacific convention on Simulated Evolution and studying, SEAL'96, held in Taejon, Korea, in November 1996. The 23 revised complete papers have been chosen for inclusion during this ebook at the foundation of two rounds of reviewing and enhancements.

Additional info for Complexity Lower Bounds using Linear Algebra (Foundations and Trends in Theoretical Computer Science)

Example text

We conclude that Volr (A) ≥ ( vr )r . On the other hand, consider the r-dimensional subspace V = v1 , . . , vr . Clearly, Rigr (A) ≤ dist(ai , V ). 3 Bounded Coefficient Complexity of Linear Transformations 51 and imagining continuing the selection for one more step, we note that dist(ai , V ) ≤ vr+1 ≤ vr . Hence, we have Rigr (A) ≤ vr . It follows that Volr (A) ≥ ( vr )r ≥ (Rigr (A))r proving the lemma. ) is a natural mathematical measure. , numbers of arbitrary value and precision if working over R or C).

Then, (1) rank(A) = r if and only if σ1 (A) ≥ · · · ≥ σr (A) > σr+1 (A) = · · · = σn (A) = 0. (2) A 2F = σ12 (A) + · · · + σn2 (A). (3) A = σ1 (A). The following important inequality [38] is often useful. 4 (Hoffman-Wielandt Inequality). Let A and B be matrices in Cn×n . Then, n [σi (A) − σi (B)]2 ≤ A − B 2 F. i=1 Hoffman and Wielandt [38] prove their result for eigenvalues of normal matrices using the Birkhoff–von Neumann characterization of doubly stochastic matrices. 3]. 2 43 Variants of Rigidity We will first consider two variants of the rigidity function based on the 2 -distance to low-rank matrices.

13), we get, (n − 2s)n/2 ≤ nr + n/2 n/2 2 . 30 Matrix Rigidity Since n k ≤ (ne/k)k , Hence, (n − 2s)n/2 ≤ 2·n/2 (nr + n/2)e n/2 . (n − 2s) ≤ ((1 + 2r)e)2 This gives us, ≤ c · r2 , for some constant c > 0. (n − c · r2 ) s≥ . 2 Since RV (r) = |C| = n · s, we have proved the theorem. 3. 17 can be improved by a constant factor with a more careful choice of the parameters. Indeed, as pointed out by Tom Hayes (private communication), by selecting √ t ≈ ( 2ern)2/3 in the proof above, it can be shown that RV (r) ≥ n2 − O(n5/3 r2/3 ).

Download PDF sample

Rated 4.43 of 5 – based on 20 votes