Many areas in science and engineering now have access to technologies that enable the rapid collection of overwhelming data volumes. While these datasets are vital for understanding phenomena from physical to biological and social systems, the sheer magnitude of the data makes even simple storage, transmission, and basic processing highly challenging. To enable efficient and accurate execution of these data processing tasks, we require new dimensionality reduction tools that 1) do not need expensive, time-consuming training, and 2) preserve the underlying geometry of the data that has the information required to understand the measured system. Specifically, the geometry to be preserved is that induced by the fact that in many applications, streaming high-dimensional data evolves on a low-dimensional attractor manifold. Importantly, we may not know the exact structure of this manifold a priori. To solve these challenges, we present randomized filtering (RF), which leverages a specific instantiation of randomized dimensionality reduction to provably preserve non-linear manifold structure in the embedded space while remaining data-independent and computationally efficient. In this work we build on the rich theoretical promise of randomized dimensionality reduction to develop RF as a real, practical approach. We introduce novel methods, analysis, and experimental verification to illuminate the practicality of RF in diverse scientific applications, including several simulated and real-data examples that showcase the tangible benefits of RF.