Achievable complexity-performance tradeoffs in lossy compression

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(·).