Exponential Strong Converse for Content Identification With Lossy Recovery

Abstract

We revisit the high-dimensional content identification with lossy recovery problem (Tuncel and Gündüz, 2014) and establish an exponential strong converse theorem. As a corollary of the exponential strong converse theorem, we derive an upper bound on the joint identification-error and excess-distortion exponent for the problem. Our main results can be specialized to the biometrical identification problem (Willems, 2003) and the content identification problem (Tuncel, 2009) since these two problems are both special cases of the content identification with lossy recovery problem. We leverage the information spectrum method introduced by Oohama and adapt the strong converse techniques therein to be applicable to the problem at hand.

Exponential Strong Converse for Successive Refinement with Causal Decoder Side Information

“…We first obtain a non-asymptotic upper bound using the information spectrum of log-likelihoods involved in the definition of ω (µ,α k ,β k ) QT (see (16)) and then apply Cramér's bound on large deviations (see e.g., [29,Lemma 13]) to obtain an exponential type non-asymptotic upper bound. Subsequently, we apply the recursive method [23] and proceed similarly as in [29] to obtain the desired result. Our method can also be used to establish similar results for other source coding problems with causal decoder side information [15], [18], [20].…”

“…i.e., strong converse holds for the k-user causal successive refinement problem. Using the strong converse theorem and Marton's change-of-measure technique [36], similarly to [29,Theorem 5], we can also derive an upper bound on the exponent of the excess-distortion probability. Furthermore, applying the one-shot techniques in [37], we can also establish a non-asymptotic achievability bound.…”

“…The proof of Lemma 2 is inspired by [23,Property 4], [29,Lemma 2] and is given in Section V. As will be shown in Theorem 3, the exponent function F (R k , D k ) is a lower bound on the exponent of the probability of non-excess-distortion probability for the k-user causal successive refinement problem. Thus, Claim (i) in Lemma 2 is crucial to establish the exponential strong converse theorem which states that the excess-distortion probability (see (3)) approaches one exponentially fast with respect to the blocklength of the source sequences.…”

“…Recall the definition of κ (α k ,β k ) (R k , D k ) in (19). Using Cramér's bound [29,Lemma 13], we obtain the following nonasymptotic exponential type upper bound on the probability of non-excess-distortion, whose proof is given in in Appendix D.…”

“…The proof of Lemma 9 is inspired by [23], [29] and given in Appendix F. In order to prove Lemma 9, we use the alternative characterizations of the rate-distortion region R in Lemma 8 and analyze the connections between the two exponent functions…”