1,757 research outputs found
On inequalities for normalized Schur functions
We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M.
Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics,
32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions}
under the usual (dominance) partial-order on partitions. We believe that our
proof technique may be helpful in obtaining similar inequalities for other
symmetric functions.Comment: This version fixes the error of the previous on
Explicit eigenvalues of certain scaled trigonometric matrices
In a very recent paper "\emph{On eigenvalues and equivalent transformation of
trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78
(2012)), the authors motivated and discussed a trigonometric matrix that arises
in the design of finite impulse response (FIR) digital filters. The eigenvalues
of this matrix shed light on the FIR filter design, so obtaining them in closed
form was investigated. Zhang \emph{et al.}\ proved that their matrix had rank-4
and they conjectured closed form expressions for its eigenvalues, leaving a
rigorous proof as an open problem. This paper studies trigonometric matrices
significantly more general than theirs, deduces their rank, and derives
closed-forms for their eigenvalues. As a corollary, it yields a short proof of
the conjectures in the aforementioned paper.Comment: 7 pages; fixed Lemma 2, tightened inequalitie
Fast projections onto mixed-norm balls with applications
Joint sparsity offers powerful structural cues for feature selection,
especially for variables that are expected to demonstrate a "grouped" behavior.
Such behavior is commonly modeled via group-lasso, multitask lasso, and related
methods where feature selection is effected via mixed-norms. Several mixed-norm
based sparse models have received substantial attention, and for some cases
efficient algorithms are also available. Surprisingly, several constrained
sparse models seem to be lacking scalable algorithms. We address this
deficiency by presenting batch and online (stochastic-gradient) optimization
methods, both of which rely on efficient projections onto mixed-norm balls. We
illustrate our methods by applying them to the multitask lasso. We conclude by
mentioning some open problems.Comment: Preprint of paper under revie
On the matrix square root via geometric optimization
This paper is triggered by the preprint "\emph{Computing Matrix Squareroot
via Non Convex Local Search}" by Jain et al.
(\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent
for computing the square root of a positive definite matrix. Contrary to claims
of~\citet{jain2015}, our experiments reveal that Newton-like methods compute
matrix square roots rapidly and reliably, even for highly ill-conditioned
matrices and without requiring commutativity. We observe that gradient-descent
converges very slowly primarily due to tiny step-sizes and ill-conditioning. We
derive an alternative first-order method based on geodesic convexity: our
method admits a transparent convergence analysis ( page), attains linear
rate, and displays reliable convergence even for rank deficient problems.
Though superior to gradient-descent, ultimately our method is also outperformed
by a well-known scaled Newton method. Nevertheless, the primary value of our
work is its conceptual value: it shows that for deriving gradient based methods
for the matrix square root, \emph{the manifold geometric view of positive
definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and
more words about the rank-deficient cas
MetaSpace II: Object and full-body tracking for interaction and navigation in social VR
MetaSpace II (MS2) is a social Virtual Reality (VR) system where multiple
users can not only see and hear but also interact with each other, grasp and
manipulate objects, walk around in space, and get tactile feedback. MS2 allows
walking in physical space by tracking each user's skeleton in real-time and
allows users to feel by employing passive haptics i.e., when users touch or
manipulate an object in the virtual world, they simultaneously also touch or
manipulate a corresponding object in the physical world. To enable these
elements in VR, MS2 creates a correspondence in spatial layout and object
placement by building the virtual world on top of a 3D scan of the real world.
Through the association between the real and virtual world, users are able to
walk freely while wearing a head-mounted device, avoid obstacles like walls and
furniture, and interact with people and objects. Most current virtual reality
(VR) environments are designed for a single user experience where interactions
with virtual objects are mediated by hand-held input devices or hand gestures.
Additionally, users are only shown a representation of their hands in VR
floating in front of the camera as seen from a first person perspective. We
believe, representing each user as a full-body avatar that is controlled by
natural movements of the person in the real world (see Figure 1d), can greatly
enhance believability and a user's sense immersion in VR.Comment: 10 pages, 9 figures. Video:
http://living.media.mit.edu/projects/metaspace-ii
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