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A higher-order neural network (HONN) can be designed to be invariant to geometric transformations such as scale, translation, and in-plane rotation. Invariances are built directly into the architecture of a HONN and do not need to be learned. Thus, for 2D object recognition, the network needs to be trained on just one view of each object class, not numerous scaled, translated, and rotated views. Because the 2D object recognition task is a component of the 3D object recognition task, built-in 2D invariance also decreases the size of the training set required for 3D object recognition. We present results for 2D object recognition both in simulation and within a robotic vision experiment and for 3D object recognition in simulation. We also compare our method to other approaches and show that HONNs have distinct advantages for position, scale, and rotation-invariant object recognition.The major drawback of HONNs is that the size of the input field is limited due to the memory required for the large number of interconnections in a fully connected network. We present partial connectivity strategies and a coarse-coding technique for overcoming this limitation and increasing the input field to that required by practical object recognition problems.