  We initiate a comprehensive experimental study of objective-based
hierarchical clustering methods on massive datasets consisting of deep
embedding vectors from computer vision and NLP applications. This includes a
large variety of image embedding (ImageNet, ImageNetV2, NaBirds), word
embedding (Twitter, Wikipedia), and sentence embedding (SST-2) vectors from
several popular recent models (e.g. ResNet, ResNext, Inception V3, SBERT). Our
study includes datasets with up to $4.5$ million entries with embedding
dimensions up to $2048$.
  In order to address the challenge of scaling up hierarchical clustering to
such large datasets we propose a new practical hierarchical clustering
algorithm B++&C. It gives a 5%/20% improvement on average for the popular
Moseley-Wang (MW) / Cohen-Addad et al. (CKMM) objectives (normalized) compared
to a wide range of classic methods and recent heuristics. We also introduce a
theoretical algorithm B2SAT&C which achieves a $0.74$-approximation for the
CKMM objective in polynomial time. This is the first substantial improvement
over the trivial $2/3$-approximation achieved by a random binary tree. Prior to
this work, the best poly-time approximation of $\approx 2/3 + 0.0004$ was due
to Charikar et al. (SODA'19).
