  Performing high level cognitive tasks requires the integration of feature
maps with drastically different structure. In Visual Question Answering (VQA)
image descriptors have spatial structures, while lexical inputs inherently
follow a temporal sequence. The recently proposed Multimodal Compact Bilinear
pooling (MCB) forms the outer products, via count-sketch approximation, of the
visual and textual representation at each spatial location. While this
procedure preserves spatial information locally, outer-products are taken
independently for each fiber of the activation tensor, and therefore do not
include spatial context. In this work, we introduce multi-dimensional sketch
({MD-sketch}), a novel extension of count-sketch to tensors. Using this new
formulation, we propose Multimodal Compact Tensor Pooling (MCT) to fully
exploit the global spatial context during bilinear pooling operations.
Contrarily to MCB, our approach preserves spatial context by directly
convolving the MD-sketch from the visual tensor features with the text vector
feature using higher order FFT. Furthermore we apply MCT incrementally at each
step of the question embedding and accumulate the multi-modal vectors with a
second LSTM layer before the final answer is chosen.
