  Topological data analysis (TDA) is gaining prominence across a wide spectrum
of machine learning tasks that spans from manifold learning to graph
classification. A pivotal technique within TDA is persistent homology (PH),
which furnishes an exclusive topological imprint of data by tracing the
evolution of latent structures as a scale parameter changes. Present PH tools
are confined to analyzing data through a single filter parameter. However, many
scenarios necessitate the consideration of multiple relevant parameters to
attain finer insights into the data. We address this issue by introducing the
Effective Multidimensional Persistence (EMP) framework. This framework empowers
the exploration of data by simultaneously varying multiple scale parameters.
The framework integrates descriptor functions into the analysis process,
yielding a highly expressive data summary. It seamlessly integrates established
single PH summaries into multidimensional counterparts like EMP Landscapes,
Silhouettes, Images, and Surfaces. These summaries represent data's
multidimensional aspects as matrices and arrays, aligning effectively with
diverse ML models. We provide theoretical guarantees and stability proofs for
EMP summaries. We demonstrate EMP's utility in graph classification tasks,
showing its effectiveness. Results reveal that EMP enhances various single PH
descriptors, outperforming cutting-edge methods on multiple benchmark datasets.
