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In this paper new general modewise Johnson-Lindenstrauss (JL) subspaceembeddings are proposed that are both considerably faster to generate andeasier to store than traditional JL embeddings when working with extremelylarge vectors and/or tensors.  Corresponding embedding results are then proven for two different types oflow-dimensional (tensor) subspaces. The first of these new subspace embeddingresults produces improved space complexity bounds for embeddings of rank-$r$tensors whose CP decompositions are contained in the span of a fixed (butunknown) set of $r$ rank-one basis tensors. In the traditional vector settingthis first result yields new and very general near-optimal oblivious subspaceembedding constructions that require fewer random bits to generate thanstandard JL embeddings when embedding subspaces of $\mathbb{C}^N$ spanned bybasis vectors with special Kronecker structure. The second result proven hereinprovides new fast JL embeddings of arbitrary $r$-dimensional subspaces$\mathcal{S} \subset \mathbb{C}^N$ which also require fewer random bits (and soare easier to store - i.e., require less space) than standard fast JL embeddingmethods in order to achieve small $\epsilon$-distortions. These new oblivioussubspace embedding results work by $(i)$ effectively folding any given vectorin $\mathcal{S}$ into a (not necessarily low-rank) tensor, and then $(ii)$embedding the resulting tensor into $\mathbb{C}^m$ for $m \leq C r \log^c(N) /\epsilon^2$.  Applications related to compression and fast compressed least squaressolution methods are also considered, including those used for fitting low-rankCP decompositions, and the proposed JL embedding results are shown to work wellnumerically in both settings.

Lower Memory Oblivious (Tensor) Subspace Embeddings with Fewer Random Bits: Modewise Methods for Least Squares