597 research outputs found
\tau-rigid modules for algebras with radical square zero
In this paper, we show that for an algebra with radical square zero
and an indecomposable -module such that is Gorenstein of
finite type or is -rigid, is -rigid if and only if the
first two projective terms of a minimal projective resolution of have no
on-zero direct summands in common. We also determined all -tilting
modules for Nakayama algebras with radical square zero. Moreover, by giving a
construction theorem we show that a basic connected radical square zero algebra
admitting a unique -tilting module is local
A Note on Gorenstein Projective Conjecture II
In this paper, we prove that Gorenstein projective conjecture is left and
right symmetric and the co-homology vanishing condition can not be reduced in
general. Moreover, the Gorenstein projective conjecture is proved to be true
for CM-finite algebras.Comment: 7pages,Section 3 is deleted and We use the notion "Auslander-Reiten
Conjecture
Classifying -tilting modules over the Auslander algebra of
We build a bijection between the set \sttilt\Lambda of isomorphism classes
of basic support -tilting modules over the Auslander algebra of
and the symmetric group , which is an
anti-isomorphism of partially ordered sets with respect to the generation order
on \sttilt\Lambda and the left order on . This restricts
to the bijection between the set \tilt\Lambda of isomorphism classes of basic
tilting -modules and the symmetric group due to
Br\"{u}stle, Hille, Ringel and R\"{o}hrle. Regarding the preprojective algebra
of Dynkin type as a factor algebra of , we show that
the tensor functor induces a bijection between
\sttilt\Lambda\to\sttilt\Gamma. This recover Mizuno's bijection
\mathfrak{S}_{n+1}\to\sttilt\Gamma for type
Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories
For an Artinian -Auslander algebra with global dimension
, we show that if admits a trivial maximal
-orthogonal subcategory of , then is a Nakayama
algebra and the projective or injective dimension of any indecomposable module
in is at most . As a result, for an Artinian Auslander
algebra with global dimension 2, if admits a trivial maximal
1-orthogonal subcategory of , then is a tilted algebra
of finite representation type. Further, for a finite-dimensional algebra
over an algebraically closed field , we show that is a
basic and connected -Auslander algebra with global dimension
admitting a trivial maximal -orthogonal subcategory of
if and only if is given by the quiver: \xymatrix{1 &
\ar[l]_{\beta_{1}} 2 & \ar[l]_{\beta_{2}} 3 & \ar[l]_{\beta_{3}} ... &
\ar[l]_{\beta_{n}} n+1} modulo the ideal generated by
. As a consequence, we get that a
finite-dimensional algebra over an algebraically closed field is an
-Auslander algebra with global dimension admitting a trivial
maximal -orthogonal subcategory if and only if it is a finite direct
product of and as above. Moreover, we give some necessary
condition for an Artinian Auslander algebra admitting a non-trivial maximal
1-orthogonal subcategory.Comment: 25 pages. This version is a combination of the orginal version of
this paper with "From Auslander Algebras to Tilted Algebras"
(arXiv:0903.0760). The latter paper has been withdraw
Tilting modules over Auslander-Gorenstein Algebras
For a finite dimensional algebra and a non-negative integer , we
characterize when the set \tilt_n\Lambda of additive equivalence classes of
tilting modules with projective dimension at most has a minimal (or
equivalently, minimum) element. This generalize results of Happel-Unger.
Moreover, for an -Gorenstein algebra with , we construct
a minimal element in \tilt_{n}\Lambda.
As a result, we give equivalent conditions for a -Gorenstein algebra to be
Iwanaga-Gorenstein. Moreover, for an -Gorenstein algebra and its
factor algebra , we show that there is a bijection between
\tilt_1\Lambda and the set \sttilt\Gamma of isomorphism classes of basic
support -tilting -modules, where is an idempotent such that
is the additive generator of projective-injective
-modules
Three results for tau-rigid modules
-rigid modules are essential in the -tilting theory introduced by
Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for
Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms
of -rigid modules. We show that every indecomposable module over iterated
tilted algebras of Dynkin type is -rigid. Finally, we give a
-tilting theorem on homological dimension which is an analog to that of
classical tilting modules.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic
Online Data Poisoning Attack
We study data poisoning attacks in the online setting where training items
arrive sequentially, and the attacker may perturb the current item to
manipulate online learning. Importantly, the attacker has no knowledge of
future training items nor the data generating distribution. We formulate online
data poisoning attack as a stochastic optimal control problem, and solve it
with model predictive control and deep reinforcement learning. We also upper
bound the suboptimality suffered by the attacker for not knowing the data
generating distribution. Experiments validate our control approach in
generating near-optimal attacks on both supervised and unsupervised learning
tasks
Adaptive Double-Exploration Tradeoff for Outlier Detection
We study a variant of the thresholding bandit problem (TBP) in the context of
outlier detection, where the objective is to identify the outliers whose
rewards are above a threshold. Distinct from the traditional TBP, the threshold
is defined as a function of the rewards of all the arms, which is motivated by
the criterion for identifying outliers. The learner needs to explore the
rewards of the arms as well as the threshold. We refer to this problem as
"double exploration for outlier detection". We construct an adaptively updated
confidence interval for the threshold, based on the estimated value of the
threshold in the previous rounds. Furthermore, by automatically trading off
exploring the individual arms and exploring the outlier threshold, we provide
an efficient algorithm in terms of the sample complexity. Experimental results
on both synthetic datasets and real-world datasets demonstrate the efficiency
of our algorithm
An Optimal Control Approach to Sequential Machine Teaching
Given a sequential learning algorithm and a target model, sequential machine
teaching aims to find the shortest training sequence to drive the learning
algorithm to the target model. We present the first principled way to find such
shortest training sequences. Our key insight is to formulate sequential machine
teaching as a time-optimal control problem. This allows us to solve sequential
teaching by leveraging key theoretical and computational tools developed over
the past 60 years in the optimal control community. Specifically, we study the
Pontryagin Maximum Principle, which yields a necessary condition for optimality
of a training sequence. We present analytic, structural, and numerical
implications of this approach on a case study with a least-squares loss
function and gradient descent learner. We compute optimal training sequences
for this problem, and although the sequences seem circuitous, we find that they
can vastly outperform the best available heuristics for generating training
sequences
Automatic Ensemble Learning for Online Influence Maximization
We consider the problem of selecting a seed set to maximize the expected
number of influenced nodes in the social network, referred to as the
\textit{influence maximization} (IM) problem. We assume that the topology of
the social network is prescribed while the influence probabilities among edges
are unknown. In order to learn the influence probabilities and simultaneously
maximize the influence spread, we consider the tradeoff between exploiting the
current estimation of the influence probabilities to ensure certain influence
spread and exploring more nodes to learn better about the influence
probabilities. The exploitation-exploration trade-off is the core issue in the
multi-armed bandit (MAB) problem. If we regard the influence spread as the
reward, then the IM problem could be reduced to the combinatorial multi-armed
bandits. At each round, the learner selects a limited number of seed nodes in
the social network, then the influence spreads over the network according to
the real influence probabilities. The learner could observe the activation
status of the edge if and only if its start node is influenced, which is
referred to as the edge-level semi-bandit feedback. Two classical bandit
algorithms including Thompson Sampling and Epsilon Greedy are used to solve
this combinatorial problem. To ensure the robustness of these two algorithms,
we use an automatic ensemble learning strategy, which combines the exploration
strategy with exploitation strategy. The ensemble algorithm is self-adaptive
regarding that the probability of each algorithm could be adjusted based on the
historical performance of the algorithm. Experimental evaluation illustrates
the effectiveness of the automatically adjusted hybridization of exploration
algorithm with exploitation algorithm
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