# Achievable complexity-performance tradeoffs in lossy compression

Achievable complexity-performance tradeoffs in lossy compression We present several results related to the complexity-performance tradeoff in lossy compression. The first result shows that for a memoryless source with rate-distortion function R(D) and a bounded distortion measure, the rate-distortion point (R(D) + γ, D + ɛ) can be achieved with constant decompression time per (separable) symbol and compression time per symbol proportional to $$\left( {{{\lambda _1 } \mathord{\left/ {\vphantom {{\lambda _1 } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \right)^{{{\lambda _2 } \mathord{\left/ {\vphantom {{\lambda _2 } {\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {\gamma ^2 }}}$$ , where λ 1 and λ 2 are source dependent constants. The second result establishes that the same point can be achieved with constant decompression time and compression time per symbol proportional to $$\left( {{{\rho _1 } \mathord{\left/ {\vphantom {{\rho _1 } \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)^{{{\rho _2 } \mathord{\left/ {\vphantom {{\rho _2 } {\varepsilon ^2 }}} \right. \kern-\nulldelimiterspace} {\varepsilon ^2 }}}$$ . These results imply, for any function g(n) that increases without bound arbitrarily slowly, the existence of a sequence of lossy compression schemes of blocklength n with O(ng(n)) compression complexity and O(n) decompression complexity that achieve the point (R(D), D) asymptotically with increasing blocklength. We also establish that if the reproduction alphabet is finite, then for any given R there exists a universal lossy compression scheme with O(ng(n)) compression complexity and O(n) decompression complexity that achieves the point (R, D(R)) asymptotically for any stationary ergodic source with distortion-rate function D(·). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Achievable complexity-performance tradeoffs in lossy compression

, Volume 48 (4) – Jan 24, 2013
24 pages

Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946012040060
Publisher site
See Article on Publisher Site

### Abstract

We present several results related to the complexity-performance tradeoff in lossy compression. The first result shows that for a memoryless source with rate-distortion function R(D) and a bounded distortion measure, the rate-distortion point (R(D) + γ, D + ɛ) can be achieved with constant decompression time per (separable) symbol and compression time per symbol proportional to $$\left( {{{\lambda _1 } \mathord{\left/ {\vphantom {{\lambda _1 } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \right)^{{{\lambda _2 } \mathord{\left/ {\vphantom {{\lambda _2 } {\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {\gamma ^2 }}}$$ , where λ 1 and λ 2 are source dependent constants. The second result establishes that the same point can be achieved with constant decompression time and compression time per symbol proportional to $$\left( {{{\rho _1 } \mathord{\left/ {\vphantom {{\rho _1 } \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)^{{{\rho _2 } \mathord{\left/ {\vphantom {{\rho _2 } {\varepsilon ^2 }}} \right. \kern-\nulldelimiterspace} {\varepsilon ^2 }}}$$ . These results imply, for any function g(n) that increases without bound arbitrarily slowly, the existence of a sequence of lossy compression schemes of blocklength n with O(ng(n)) compression complexity and O(n) decompression complexity that achieve the point (R(D), D) asymptotically with increasing blocklength. We also establish that if the reproduction alphabet is finite, then for any given R there exists a universal lossy compression scheme with O(ng(n)) compression complexity and O(n) decompression complexity that achieves the point (R, D(R)) asymptotically for any stationary ergodic source with distortion-rate function D(·).

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jan 24, 2013

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