arXiv research

A novel circuit motif uses sister cells for inference with correlated priors.

Sparse Convex Biclustering improves accuracy and robustness in high-dimensional datasets.

We prove a concordance analogue of Gabai's $4$-dimensional light bulb theorem. That is, we show that when $R$ and $R'$ are homotopically (smoothly) embedded $2$-spheres in a $4$-manifold $X^4$ where $π_1(X^4)$ has no $2$-torsion and one of $R$ or $R'$ has a transverse sphere, then $R$ and $R'$ are concordant. When $π_1…

4D theorem for disks, generalizing previous work.

Evaluating, explaining, and visualizing high-level concepts in generative models, such as variational autoencoders (VAEs), is challenging in part due to a lack of known prediction classes that are required to generate saliency maps in supervised learning. While saliency maps may help identify relevant features (e.g., p…

New smoothing techniques for topological surfaces in 4-manifolds.

New 4-manifold examples show necessary conditions for 4D Light Bulb Theorem.

In this note I present my understanding of, that is to say the way I look at, David Gabai's proof of his recent 4-Dimensional Light Bulb Theorem (4D-LBT). His construction, entirely smooth, is an ingenious amalgam of classical moves, and represents the first new hands-on advance in constructive smooth 4-manifold theory…

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple close…

The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.

Binary autoencoder with sparse hidden layer preserves information and zero reconstruction error.

In this note, we combine the recent 4-dimensional light bulb theorem of David Gabai and a recent construction of concordances for knots in $S^2\times S^1$ due to Eylem Zeliha Yildiz to construct a concordance between the standard surface of genus $g$ in $S^2\times S^2$ and any homologous surface.

We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Ki…

Machine learning (ML) classifiers always benefit from more informative input features. We seek to auto-generate stronger feature sets in order to address the difficulty that ML methods often experience given limited training data. A wide range of biological neural nets (BNNs) excel at fast learning, implying that they …

If $Σ$ and $Σ'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance bet…

Link framings can only change when a 3-manifold has a non-separating sphere.

We prove a concordance version of the 4-dimensional light bulb theorem for $π_1$-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if $F_0$ and $F_1$ are such surfaces in a 4-manifold $X$ that are homotopic and there exists an immersed framed…

New algorithm uses random projections for robust, sparse data classification.

Predicting the relationship between a molecule's structure and its odor remains a difficult, decades-old task. This problem, termed quantitative structure-odor relationship (QSOR) modeling, is an important challenge in chemistry, impacting human nutrition, manufacture of synthetic fragrance, the environment, and sensor…

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" fo…

We propose a new class of data-independent locality-sensitive hashing (LSH) algorithms based on the fruit fly olfactory circuit. The fundamental difference of this approach is that, instead of assigning hashes as dense points in a low dimensional space, hashes are assigned in a high dimensional space, which enhances th…

New method for classifying disk embeddings in 4-manifolds.

New phenomena in 4-manifolds show discs with special properties.

The fruit fly Drosophila's olfactory circuit has inspired a new locality sensitive hashing (LSH) algorithm, FlyHash. In contrast with classical LSH algorithms that produce low dimensional hash codes, FlyHash produces sparse high-dimensional hash codes and has also been shown to have superior empirical performance compa…

A new classifier encodes local neighborhoods for each class using Fly Bloom Filters.

The paper simplifies embedding spaces in manifolds by attaching handles.

The mushroom body is the key network for the representation of learned olfactory stimuli in Drosophila and insects. The sparse activity of Kenyon cells, the principal neurons in the mushroom body, plays a key role in the learned classification of different odours. In the specific case of the fruit fly, the sparseness o…

New algorithm reduces runtime for robust sparse mean estimation.

The paper solves isotopy problems on 4-manifolds and classifies symplectic structures.

We consider the problem of enumerating relevant features hidden in other irrelevant information for multi-labeled data, which is formalized as learning juntas. A $k$-junta function is a function which depends on only $k$ coordinates of the input. For relatively small $k$ w.r.t. the input size $n$, learning $k$-junta fu…

Method reconstructs aneurysm growth history from patient parameters using physics-informed autoencoder.

Detects exotic surfaces without smooth invariants, providing first example of knotted RP².