It suggests that, for any m { k,...,n 1 } and z R, L A.2 Proofs for Lemma 2 and 3 for the case when K is unknown in 4 Lemma 2 . It suggests that, for any m { 0,...,n 1 } and z R, L For any m { 0,...,n 1 } and z R, we have L A.3 Additional tricks for methods proposed in 3. Finding optimal CP vector when z = in paraCP(n,k, ˆ T Additional pruning condition for parametric DP when K is fixed. In 3.3, we showed that Lemma 4. F orn [ N ], and k [ K ], let T Therefore, it fails to control the false positive rate. This is asymptotic test for multiple detected CPs. Fused Lasso (proposed by the same authors), is worse than BinSeg-SI. BinSeg-SI had been considered as a computationally efficient approximation of the problem in (7), where the authors additionally condition on extra information for computational tractability, e.g., the order that CPs are detected.

Changepoint (CP) detection is a fundamental problem and has been studied in many areas.

This dataset includes images from homes with no internet access and incomes as low as $26.99 per

Existing bias mitigation methods for DNN models primarily work on learning debi-ased encoders.

The first column is the input RGB images.

Plugging back in (11) yields the ideal result.

For the convenience of the reader, we summarize a list of notations blow. 1. null G In Appendix B.1, we present a general statement of Theorem 3.1 (a) along with its proof. Theorem 3.1 (a) states the order recovery guarantee for a specified parameter We summarize the bounds for (I) and (II) in Lemma B.1 and Lemma B.2, which can be found in Collecting the results in Lemma B.1 and Lemma B.2 and reorganizing the terms in the inequalities, we have the following conclusion. We now state the proof of this Lemma. Then we bound the first term using the concentration bound on Chi-squared random variables. For the non-identifiable models, we can use Lemma H.1 in a similar way to obtain that with probability We now state the proof of this Lemma.