36,387 research outputs found
ΠΠΎΠ²ΡΠ΅ ΠΊΠ»Π΅ΡΠΎΡΠ½ΡΠ΅ ΠΌΠ΅ΡΠΎΠ΄Ρ ΡΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΡ ΠΌΠ°ΡΡΠΈΡ
ΠΠ°ΠΏΡΠΎΠΏΠΎΠ½ΠΎΠ²Π°Π½ΠΎ Π΄Π²Π° Π½ΠΎΠ²ΠΈΡ
ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΈ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΠΌΠ°ΡΡΠΈΡΡ, ΡΠΊΡ Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡΡΡ ΠΎΡΡΠΈΠΌΠ°ΡΠΈ ΠΊΠ»ΡΡΠΈΠ½Π½Ρ Π°Π½Π°Π»ΠΎΠ³ΠΈ Π²ΡΠ΄ΠΎΠΌΠΈΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΡΠ² ΠΌΠ°ΡΡΠΈΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ Π·Ρ Π·ΠΌΠ΅Π½ΡΠ΅Π½ΠΎΡ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΡΡΡΡ, ΠΏΠΎΡΡΠ²Π½ΡΠ½ΠΎ Π· Π°Π½Π°Π»ΠΎΠ³Π°ΠΌΠΈ, ΠΎΡΡΠΈΠΌΠ°Π½ΠΈΠΌΠΈ Π½Π° ΠΎΡΠ½ΠΎΠ²Ρ Π²ΡΠ΄ΠΎΠΌΠΈΡ
ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΡΠ² ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΠΌΠ°ΡΡΠΈΡΡ. ΠΠΎΠ²ΠΈΠΉ ΡΠ²ΠΈΠ΄ΠΊΠΈΠΉ ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ Π΄ΠΎΠ·Π²ΠΎΠ»ΡΡ ΠΌΡΠ½ΡΠΌΡΠ·ΡΠ²Π°ΡΠΈ Π½Π° 15% ΠΌΡΠ»ΡΡΠΈΠΏΠ»ΡΠΊΠ°ΡΠΈΠ²Π½Ρ, Π°Π΄ΠΈΡΠΈΠ²Π½Ρ Ρ Π·Π°Π³Π°Π»ΡΠ½Ρ ΡΠΊΠ»Π°Π΄Π½ΡΡΡΡ Π²ΡΠ΄ΠΎΠΌΠΈΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΡΠ² ΠΌΠ°ΡΡΠΈΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ. ΠΠΎΠ²ΠΈΠΉ Π·ΠΌΡΡΠ°Π½ΠΈΠΉ ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΡΠ΄Π½ΡΡ ΠΌΠ΅ΡΠΎΠ΄ ΠΠ΅ΠΉΠ΄Π΅ΡΠΌΠ°Π½Π° ΡΠ· Π·Π°ΠΏΡΠΎΠΏΠΎΠ½ΠΎΠ²Π°Π½ΠΈΠΌ ΡΠ²ΠΈΠ΄ΠΊΠΈΠΌ ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΠΌ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠΌ, ΡΠΎ ΠΏΡΠΈΠ·Π²ΠΎΠ΄ΠΈΡΡ Π΄ΠΎ ΠΌΡΠ½ΡΠΌΡΠ·Π°ΡΡΡ Π½Π° 28% ΠΌΡΠ»ΡΡΠΈΠΏΠ»ΡΠΊΠ°ΡΠΈΠ²Π½ΠΎΡ, Π°Π΄ΠΈΡΠΈΠ²Π½ΠΎΡ Ρ Π·Π°Π³Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΠΎΡΡΡ Π·Π°Π·Π½Π°ΡΠ΅Π½ΠΈΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΡΠ². ΠΡΡΠ½ΠΊΠΈ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΠΎΡΡΡ ΡΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΡΠ² ΠΏΠΎΠ΄Π°Π½ΠΎ Π½Π° ΠΏΡΠΈΠΊΠ»Π°Π΄Ρ ΠΎΡΡΠΈΠΌΠ°Π½Π½Ρ ΠΊΠ»ΡΡΠΈΠ½Π½ΠΈΡ
Π°Π½Π°Π»ΠΎΠ³ΡΠ² ΡΡΠ°Π΄ΠΈΡΡΠΉΠ½ΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΠΌΠ°ΡΡΠΈΡΡ.The paper proposes two new cellular methods of matrix multiplication, which allow obtaining cellular analogs of the well-known matrix multiplication algorithms with reduced computational complexity, as compared with the analogs derived on the basis of the well-known cellular methods of matrix multiplication. The new fast cellular method reduces by 15% the multiplicative, additive, and overall complexities of the mentioned algorithms. The new mixed cellular method combines the Laderman method with the proposed fast cellular method. The interaction of these methods reduces by 28% the multiplicative, additive, and overall complexities of the matrix multiplication algorithms. The computational complexity of these methods are estimated using the model of getting cellular analogs of the traditional matrix multiplication algorithm
Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation
We study the compressed sensing (CS) signal estimation problem where an input
signal is measured via a linear matrix multiplication under additive noise.
While this setup usually assumes sparsity or compressibility in the input
signal during recovery, the signal structure that can be leveraged is often not
known a priori. In this paper, we consider universal CS recovery, where the
statistics of a stationary ergodic signal source are estimated simultaneously
with the signal itself. Inspired by Kolmogorov complexity and minimum
description length, we focus on a maximum a posteriori (MAP) estimation
framework that leverages universal priors to match the complexity of the
source. Our framework can also be applied to general linear inverse problems
where more measurements than in CS might be needed. We provide theoretical
results that support the algorithmic feasibility of universal MAP estimation
using a Markov chain Monte Carlo implementation, which is computationally
challenging. We incorporate some techniques to accelerate the algorithm while
providing comparable and in many cases better reconstruction quality than
existing algorithms. Experimental results show the promise of universality in
CS, particularly for low-complexity sources that do not exhibit standard
sparsity or compressibility.Comment: 29 pages, 8 figure
SwiftFormer: Efficient Additive Attention for Transformer-based Real-time Mobile Vision Applications
Self-attention has become a defacto choice for capturing global context in
various vision applications. However, its quadratic computational complexity
with respect to image resolution limits its use in real-time applications,
especially for deployment on resource-constrained mobile devices. Although
hybrid approaches have been proposed to combine the advantages of convolutions
and self-attention for a better speed-accuracy trade-off, the expensive matrix
multiplication operations in self-attention remain a bottleneck. In this work,
we introduce a novel efficient additive attention mechanism that effectively
replaces the quadratic matrix multiplication operations with linear
element-wise multiplications. Our design shows that the key-value interaction
can be replaced with a linear layer without sacrificing any accuracy. Unlike
previous state-of-the-art methods, our efficient formulation of self-attention
enables its usage at all stages of the network. Using our proposed efficient
additive attention, we build a series of models called "SwiftFormer" which
achieves state-of-the-art performance in terms of both accuracy and mobile
inference speed. Our small variant achieves 78.5% top-1 ImageNet-1K accuracy
with only 0.8 ms latency on iPhone 14, which is more accurate and 2x faster
compared to MobileViT-v2. Code: https://github.com/Amshaker/SwiftFormerComment: Technical repor
Time for dithering: fast and quantized random embeddings via the restricted isometry property
Recently, many works have focused on the characterization of non-linear
dimensionality reduction methods obtained by quantizing linear embeddings,
e.g., to reach fast processing time, efficient data compression procedures,
novel geometry-preserving embeddings or to estimate the information/bits stored
in this reduced data representation. In this work, we prove that many linear
maps known to respect the restricted isometry property (RIP) can induce a
quantized random embedding with controllable multiplicative and additive
distortions with respect to the pairwise distances of the data points beings
considered. In other words, linear matrices having fast matrix-vector
multiplication algorithms (e.g., based on partial Fourier ensembles or on the
adjacency matrix of unbalanced expanders) can be readily used in the definition
of fast quantized embeddings with small distortions. This implication is made
possible by applying right after the linear map an additive and random "dither"
that stabilizes the impact of the uniform scalar quantization operator applied
afterwards. For different categories of RIP matrices, i.e., for different
linear embeddings of a metric space
in with , we derive upper bounds on the
additive distortion induced by quantization, showing that it decays either when
the embedding dimension increases or when the distance of a pair of
embedded vectors in decreases. Finally, we develop a novel
"bi-dithered" quantization scheme, which allows for a reduced distortion that
decreases when the embedding dimension grows and independently of the
considered pair of vectors.Comment: Keywords: random projections, non-linear embeddings, quantization,
dither, restricted isometry property, dimensionality reduction, compressive
sensing, low-complexity signal models, fast and structured sensing matrices,
quantized rank-one projections (31 pages
ΠΠΎΠ²ΡΠ΅ Π±ΡΡΡΡΡΠ΅ Π³ΠΈΠ±ΡΠΈΠ΄Π½ΡΠ΅ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ ΡΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΡ ΠΌΠ°ΡΡΠΈΡ
ΠΠ°ΠΏΡΠΎΠΏΠΎΠ½ΠΎΠ²Π°Π½ΠΎ Π½ΠΎΠ²i Π³iΠ±ΡΠΈΠ΄Π½i Π°Π»Π³ΠΎΡΠΈΡΠΌΠΈ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ (n x n)-ΠΌΠ°ΡΡΠΈΡΡ, ΠΏΡΠΈ ΠΏΠΎΠ±ΡΠ΄ΠΎΠ²Ρ ΡΠΊΠΈΡ
Π²ΠΈΠΊΠΎΡΠΈΡΡΠ°Π½ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΠ΅ΠΉΠ΄Π΅ΡΠΌΠ°Π½Π° Π΄Π»Ρ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ (3 x 3)-ΠΌΠ°ΡΡΠΈΡΡ. ΠΠΎΡΡΠ²Π½ΡΠ½ΠΎ Π· Π²ΡΠ΄ΠΎΠΌΠΈΠΌΠΈ Π³ΡΠ±ΡΠΈΠ΄Π½ΠΈΠΌΠΈ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°ΠΌΠΈ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΠΌΠ°ΡΡΠΈΡΡ Π½ΠΎΠ²Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΠΈ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·ΡΡΡΡΡΡ ΠΌΡΠ½ΡΠΌΡΠ·ΠΎΠ²Π°Π½ΠΎΡ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΡΡΡΡ. ΠΠ°Π²Π΅Π΄Π΅Π½ΠΎ ΠΎΡΡΠ½ΠΊΠΈ ΠΌΡΠ»ΡΡΠΈΠΏΠ»ΡΠΊΠ°ΡΠΈΠ²Π½ΠΎΡ, Π°Π΄ΠΈΡΠΈΠ²Π½ΠΎΡ ΡΠ° Π·Π°Π³Π°Π»ΡΠ½ΠΎΡ ΡΠΊΠ»Π°Π΄Π½ΠΎΡΡΡ Π² ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°Ρ
.New hybrid algorithms are proposed for multiplying (n x n)-matrices. They are based on Ladermanβs algorithm for multiplying (3 x 3)-matrices. As compared with the well-known hybrid matrix multiplication algorithms, the new algorithms are characterized by the minimum computational complexity. The multiplicative, additive, and overall complexities of the algorithms are estimated
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a
2x2-block-matrix by its transpose using 5 block products (3 recursive calls and
2 general products) over C or any finite field.We use geometric considerations
on the space of bilinear forms describing 2x2 matrix products to obtain this
algorithm and we show how to reduce the number of involved additions.The
resulting algorithm for arbitrary dimensions is a reduction of multiplication
of a matrix by its transpose to general matrix product, improving by a constant
factor previously known reductions.Finally we propose schedules with low memory
footprint that support a fast and memory efficient practical implementation
over a finite field.To conclude, we show how to use our result in LDLT
factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec
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